Skyrim:Sneak

The Sneak skill governs your ability to escape detection by others through reducing your visibility and noisiness. It is primarily useful for committing crimes without being witnessed, for evading confrontation with enemies, and for executing surprise attacks that inflict critical damage.  
  
Perks in the Sneak skill tree increase your chances of escaping detection, increase the damage you inflict with sneak attacks, and allow you to avoid setting off traps using pressure plate triggers. They can also give you the ability to perform a forward roll while sneaking and even briefly disappear from sight in the middle of combat. The Sneak skill tree has a total of 9 perks, requiring a total of 13 perk points to fill.

In-game Description: Sneaking is the art of moving unseen and unheard. Highly skilled sneaks can even hide in plain sight.    

Skill Perks
 
 
 

 

Skill Usage
Toggling the sneak control causes your character to crouch or stand, indicating that you have entered or left sneak mode. 

While sneaking, the crosshair becomes a special reticle indicating the following statuses:
A flat horizontal line means you are completely hidden. 
A slightly open eye means an enemy thinks a target may be near.
A half-open eye means the enemy is actively searching for you.
An open eye means the enemy knows where you are or can see you.  

"DETECTED" will flash under the reticle if an enemy has spotted you and is moving to attack you, while the word "HIDDEN" flashes once all enemies that had previously been alerted are no longer aware of your presence. 

Remaining Undetected
Remaining undetected is useful for getting into position to initiate a sneak attack or to commit a crime unwitnessed.  You have a "detection value"; how this is used is very involved - there's an entire system of Stealth Points, like hit points but for stealth - but you can think of it as how easy you are to detect: higher is easier.

Detection Value = fSneakBaseValue + (Sound factor * Distance attenuation ) + (Visual factor * Distance attenuation) + (Noticer skill factor * Distance attenuation) + (Noticer skill factor - Sneaker skill factor)

Replacing with the constants used in the unmodded game:

fSneakBaseValue = -15
Distance Attenuation = ((fSneakMaxDistance - distance) / fSneakMaxDistance) 
fSneakMaxDistance = 2500
fSneakDistanceAttenuationExponent = 2
Distance Attenuation = ((2500 - distance)/2500) 
Distance Attenuation = (1 - distance/2500) 

Detection Value = -15 + (Sound factor + Visual factor + Noticer skill factor) * (1 - distance/2500)  + (Noticer skill factor - Sneaker Skill factor)

As you can see, the Sneak skill - and hence its Stealth perks - does not reduce your sound factor or visual factor; it reduces the Noticer's skill factor.

Your Sneak skill level determines your base chance of remaining undetected, though this chance is significantly modified by a number of other factors, potentially to the point of making you completely undetectable:
The Stealth perk and several other perks on the Sneak skill tree.
Factors influencing visibility:
Line of Sight: All seeing creatures and NPCs have a defined cone of vision; if you fall within it, you are more likely to be detected.
Light: you are more visible in brighter light; this modifier carries significant effects across the entire range of light levels.
The glow of light over your hands when you have a spell equipped makes you easier to detect, but the aura given from enchantments does not.
Invisibility: being invisible is obviously a powerful aid to sneaking, but does not influence enemies' ability to hear you.
Noise Level. Factors influencing the amount of noise you make are:
Movement Speed: running is noisier than walking. Moving very slowly generally makes little noise. Extra noise from running is negated by the Silence perk.
Armor: moving in armor makes more noise; the combined weight of all equipped armor is correlated with its volume.
Muffle: A magical effect that reduces or eliminates noise.
Spells: casting a spell makes noise unless you have the Quiet Casting perk. Summon spells and most dragon shouts are particularly loud.
Weapons: swinging a weapon larger than a dagger makes noise.
The noise you hear when you unsheathe a weapon is not heard by enemies.
Previous detection: An enemy that was previously alerted to your presence (even if your reticle indicated only a slightly open eye) will remember this, and is more likely to spot you again compared to an enemy who had not yet been alerted to your presence.

Sound Factor
This is fSneakSoundsMult * (Movement + Action) = (Movement + Action), as fSneakSoundsMult is 1 in the unmodded game.  If the Sneaker does not have line-of-sight (LoS) to the Noticer, this is then multiplied by fSneakSoundLosMult, which is 0.3:

Sound Factor = (Movement + Action)*(1 if Sneaker has LoS to Noticer, 0.3 if not)

Movement:
(fSneakEquippedWeightBase + fSneakEquippedWeightMult * armor weight)*(fSneakRunningMult if running)
fSneakEquippedWeightBase = 12 (everyone counts as wearing 12 weight before actually equipping anything)
fSneakEquippedWeightMult = 0.5
fSneakRunningMult = 2 (you are twice as easy to hear while running)
Movement = (12 + (armor weight)/2)*(2 if Running)*Muffle
This value is simply set to 0 when not moving, and the muffle effect applies to the entire value if you have it from somewhere - for example, the Muffled Movement perk provides 50% muffle (0.5), and the Silence perk provides 100% muffle (0.0).
Action:
ActionSound * fSneakActionMult
fSneakActionMult = 2
ActionSound varies by action, but the various sounds include spells and shouts (if you don't have Quiet Casting), and attacking with (not drawing or sheathing) any weapon that is not a dagger.
Action = ActionSound * 2, 

Total:
Sound Factor = ((12 + (armor weight)/2)*(2 if Running)*Muffle + 2*ActionSound)*(1 if Sneaker has LoS to Noticer, 0.3 if not)

Sneak Attacks
You perform a sneak attack when you physically strike someone who is unaware of your presence. The amount of damage you inflict via sneak attack is generally increased by a factor that depends on the weapon type and whether you have unlocked certain perks.



If you equip the Shrouded Gloves, Shrouded Hand Wraps, Jester's Gloves, or Cicero's Gloves (from the Dark Brotherhood questline), the sneak attack bonus for the Backstab and ''Assassin's Blade'' perks are doubled (x12 and x30).
The sneak multiplier is only applied to the physical damage done by a weapon, i.e. the damage shown for the weapon in your inventory.
Damage increases from Smithing improvements do benefit from the sneak attack multipliers.
Poison and enchantment-related damage enhancements are added to damage inflicted by power attacks, their effects are not increased by performing sneak attacks
Both bow and arrow damage are multiplied.
Dual-wielded weapons and power attacks are multiplied.  
Lunging power attacks (hold attack while pressing forward) from sneak mode are especially powerful, although if mistimed, they can alert an enemy before the strike can land (and thus losing the bonus damage).

The very acts involved in preparing for and attacking can result enemies detecting you, and when this occurs the damage you deal will receive no sneak attack bonus. One of the reasons for this is that attacking with weapons (except daggers) itself creates noise. To minimize this, maximize all other factors that enhance your probability of remaining undetected: attack from behind, stay in the shadows, and ensure that the sneak reticle is flat (you are hidden), etc.

If a sneak attack does not kill an enemy, you will become more easily detectable, and your victim will usually start looking for you.  However, if you are not immediately detected (if the sneak "eye" is not yet fully open), you can sometimes quickly land a second successful sneak attack. Enemies who witness the attack (by sight or by sound) are also likely to be alerted to your presence, even if the victim was killed. When using a bow to perform sneak attacks on multiple enemies, position yourself such that after killing one of them you can sneakily relocate behind nearby cover. If you are not detected or become detected but hidden again, you may be able to continue with another sneak attack.

Followers
NPC followers will automatically enter sneak mode whenever you do. However, you may be better off having any follower wait out of range of enemies' sight and hearing: You will be stealthier and in greater control operating solo as a stealth style fighter. Followers' chances of being detected are calculated separately from yours, based on their sneak skill and armor. Even followers who have relatively better sneak skill levels are generally more likely than you to be detected. 

Further, if detected, followers are very likely to storm in and attack, reducing your tactical control over a situation.  

If you are sneaking with a follower at night, ensure that they are not carrying any torches, as they will reliably equip and light them while sneaking, obviously a poor idea in general.

Dogs bark, making them very poor followers if you wish to be sneaky.

There are many quests which give you temporary followers who&mdash;unlike regular followers&mdash;make your status "visible" (open eye) if they can see you, which may ruin your sneak attack bonus against enemies. Also, activating them while sneaking will attempt a pickpocket rather than start a conversation. Finally, they may make nervous comments if you are sneaking near them, e.g. "I thought I heard something", as if they were hostile NPCs.

Skill Increases
Character Creation
The following races have an initial skill bonus to Sneak:
+10 bonus: Khajiit
+5 bonus: Argonian, Bosmer, Dunmer

Trainers
Khayla with the Khajiit Caravans (Common)
Garvey in Markarth (Expert)
Delvin Mallory of the Thieves Guild in Riften (Master)

Skill Books
2920, Last Seed, v8
Legend of Krately House
The Red Kitchen Reader
Sacred Witness
Three Thieves

Free Skill Boosts
+1 Sneak reward from Stands-In-Shallows (Windhelm) for completing the quest Some Light Theft
+1 Sneak reward from Noster Eagle-Eye (Solitude) for completing the quest Dungeon Delving (Caves).
+1 Sneak reward (as well as +1 to all other stealth skills) from Inge Six Fingers (Bards College) for completing the quest Finn's Lute.
+5 Sneak reward (as well as +5 to all other stealth skills) by selecting "The Path of Shadow" from the Oghma Infinium after completing the quest Discerning the Transmundane.

Gaining Skill XP
Successful sneak attacks give the biggest skill increases. Melee sneak attacks give 12x as much as ranged sneak attacks. The amount is flat per sneak attack and does not depend on the damage done. 
Releasing the Calm spell in your left hand immediatly after a sneak attack with your right hand will enable you to to gain XP quickly. Using a low damage weapon or a punch will preserve the target's health. Trolls are good targets due to their health regeneration. Otherwise you need to heal the target with a spell or wait until they have regained their health.
If dual-wielding, a sneak power attack can potentially provide up to three times as much experience, since each of its three strikes counts as a separate stealth attack. This assumes all three attacks land, while remaining undetected, and that the target isn't killed by the first or second blow. Thus it only applies to more powerful targets.
Good 'dummies' for training with sneak attacks include: Balgruuf the Greater (After you return the Dragonstone, but before you head to Western Watchtower), Shadowmere, Hadvar/Ralof during Unbound, conjured creatures, especially Dremora Lords as they always appear facing away from you and won't turn around without extended provocation, and the Greybeards while praying. Sleeping enemies are also easy to sneak up on and backstab.
Simply being in stealth mode around any NPC increases the Sneak skill very slowly. It is somewhat faster when the NPC is hostile or owns items nearby (e.g., shopkeepers), however. The sickly soldier in the Temple of Kynareth in Whiterun is an easy target as he does not move. A dead thrall can be effective as well. The experience gain is proportional to the number of NPCs nearby. The outside environs of the Goldenglow Estate offer an excellent opportunity to increase stealth. As many as 9 NPCs, all actively hostile, are in the same area. Cautiously creeping back and forth can raise sneak a level every few minutes.
When your hands are unbound by either Ralof or Hadvar, before using the Helgen Keep key, it is possible to pick up a dagger from a dead imperial soldier, and sneak attack them repeatedly, to quickly level up sneak.

Notes
Some Oblivion-Skyrim differences in terms of the sneak skill: 
Pickpocket has been split into a separate skill from Sneak.
In Skyrim, unlike Oblivion, your Sneak skill can increase even if you are stationary. If the enemy is aware of you (eye partially open) you will level up the skill faster when they lose track of you (hidden).
 A simple way to train the Sneak skill is to autorun/walk against the cave wall near the sleeping bear in the tutorial. In this way you can train your sneak skill to any level to the maximum 100 while away from your keyboard. (You can use a rubber band for consoles)
Your weapons are swung slower when in sneak mode.  Be sure to stand back up after being detected, as remaining in sneak mode by that point is only a liability.
It appears that obtaining the Block perk Block Runner allows you to move at normal speed while sneaking and blocking. It is not clear whether this was intended in development.
Similarly, rapidly switching spells on your hotkeys (1-8) increases your movement speed while sneaking.
The level 100 Sneak perk turns the user invisible for a short moment upon activation. It will override and terminate any ongoing invisibility effects, such as the Shadowcloak of Nocturnal ability, making the user visible almost immediately after sneaking. However, it won't cancel the Illusion Invisibility spell. Since it turns you invisible for a fraction of a second, it's actually possible to crouch in front of an enemy who's already seen you and land a sneak attack.
The Silent Roll ability can be useful for more than just stealth. In some cases it's possible to use the roll to go up slopes that are difficult or impossible to climb other ways. (Horses can climb steeper slopes, but where you can't have or don't want one, it's a viable alternative.)
It is possible to carry out multiple rolls in several ways:  
In third-person view, it simply works correctly.
In first-person view, both hands need to have a weapon or shield in them, including two-handed weapons and bows.
Failing that, i.e. if you have a spell in your left hand, if you equip your weapon after a complete roll it will cause you to roll again. Sheathing your weapon after the next complete roll will cause you to roll again.
It is possible to gain Sneak skill experience very easily by sneak attacking people and creatures who will not immediately become hostile. Note, however, that many of these people will become hostile if you attempt to pickpocket them at the same time and fail. Good targets include, but are not limited to:
Any of the Greybeards during their teaching of the Unrelenting Force. They will not become hostile, and also have a large quantity of health.
Either Hadvar or Ralof during the opening quest Unbound.
Ancano at the College of Winterhold, prior to completing Revealing the Unseen, as he will be standing alone in front of the Eye of Magnus and will not become hostile. Note that if you attempt to pickpocket Ancano and fail, he will likely become hostile.
Followers, who can be told to wait before sneak attacking them. Outside of combat, their health regenerates very quickly and they will not become hostile as long as they are your follower but they will leave your service if you attack too many times in succession. Wait for them to finish telling you off before performing another sneak attack.
Your summoned or reanimated creatures and NPCs. Note that, like followers, these will become hostile with enough attacks in quick succession. However, triggering hostility is necessary to gain conjuration xp for the spells used.
Additionally, using this method with one handed weapons and Healing Hands, both One-handed and Restoration can be leveled up simultaneously by constantly healing the target while striking them whenever they become unaware of your presence.
Guards will comment negatively if you have a high sneak skill and say, "Hands to yourself, sneak thief."
It is possible to roll before the perk is unlocked. Simply jump (while moving forward) and as you are falling press and hold the sprint button.
Pacify effects like the Calm spell, if successful, will allow you to sneak again after you've been detected.

Bugs
 
This can also be done by activating something which triggers third-person mode (e.g. Ore Vein, Grindstone) while in Sneak mode. When the player returns to first-person mode, they will be able to perform the forward roll despite the reticle indicating they are not in Sneak mode. This glitch is fixed by switching back to Sneak mode or third-person view.
This can happen when jumping while stealthed, then holding down the sprint key and forward movement key when landing; you will execute a single combat-roll upon landing.
 
 
 
 
This also occurs when activating any crafting apparatus (Forge, Alchemy Table, Arcane Enchanter, Cooking Pot) while sneaking. Upon exiting, in first-person mode your character's view will be lower as if sneaking, and you will move slower. However, no sneaking reticle appears and you cannot Pickpocket (even without the perk, it is possible to do the silent roll; however, it is quite awkward and doesn't always work). In third-person mode you will be standing and can sprint. This is fixed the same way as above.
Unsneaking during a roll allows the player to input more roll commands during the roll, each one restarting the roll animation.