I'm in the US, and I did a lot of research on banks and checking accounts about a year ago when I was trying to find the best setup (for me).

I hardly ever go into a bank branch, so I was willing to sacrifice having that possibility if it meant lower fees and less service.

I found four good possibilities, and I would recommend any of these to others if you can get them (drawbacks/caveats in parenthesis):

1. Charles Schwab (lowest limit on mobile check deposits, was $2000/day max when I checked)
2. USAA (available to family members of military only when I checked)
3. ING Direct (later bought by Capital One and renamed to Capital One 360 - this is what I now use)
4. Simple (formerly banksimple, now just simple.com)

I was on the waiting list for Simple, but before they gave me the invite I ended up getting set up with ING Direct, and even though they are now Capital One 360, nothing about the program has changed and it meets all my needs - it's great! The best parts, besides the no monthly fees etc which was a prerequisite for me, is they give you a small credit line attached to your primary account ($500 to start though you can ask for an increase if you want), and if you do happen to overdraft they just charge you interest based on the overdraft amount, which ends up being a few pennies at the end of the month - no overdraft fees at all. If you don't have the credit line or you go over, the transactions are just declined.

I still have a Charles Schwab account, because in addition to no monthly fees, they refund ALL atm fees from anywhere, even the ones that the atm owners charge. You can literally use a $10-fee atm at like disneyland or on a cruise ship, and they will refund the fee. I keep a small amount of cash in here for when I want to get cash from an ATM and don't want to look for one that is feeless for my regular bank (Capital One 360 which uses AllPoint atms so they are plentiful, but hey my laziness knows no bounds)

I was never able to try USAA despite hearing great things because no one I know was in the military.

I was planning to switch to Simple once I was able, but ended up staying with Capital One 360 since it's great. Oh yeah, another awesome feature you don't get anywhere else, that started with ING direct but capital one has kept it, is you can open an unlimited amount of savings accounts. It's really one savings account in the background, but they treat the amounts separately as if they are separate accounts, so from the front end it seems that they are. This is awesome for setting up savings goal, automatic savings deductions for various purposes (like quarterly tax estimates if you are self-employed), etc.

The stupid employer I'm at wouldn't let me sign up for direct deposit until I had been working here for 6 months (I know, what kind of policy is that, right??) -- and even though it's the highest mobile check deposit limit I could find, Capital One 360 still has a $3000-per-check limit on mobile deposit. My employer not only wouldn't let me get direct deposit right away, but pays monthly, so since my paycheck was bigger than the limit I was forced to go back to a "regular" bank for a few months.

So I got a Bank of America account. They charged me $12 a month for the privilege of using their services without having direct deposit set up (if I could set it up I wouldn't need them!). Closed that account as soon as I was able to get direct deposit, which I set up for my CO-360 account. Branch manager of BoA was disappointed (ha).

I did move to a new apartment in the middle of that, and needed a cashier's check or money order for the deposit. Years past had always just gotten a cashier's check from my bank, they usually let you do like one a month for free if you had an account -- nope they don't do that anymore. $10 fee. (Screw that, went to Walmart and got a money order for 49 cents).
Also asked about notarization -- don't need it right now but since I was asking about cashier's checks anyway thought I'd ask if I could still get something notarized if I had an account -- nope sorry, they do have a notary but they are only allowed to notarize official Bank of America documents. Wtf??

Anyway, my BofA account is closed as of a few weeks ago, and I couldn't be happier to not be their customer anymore. Fully set up with direct deposit to Capital One 360 now, and that's where I intend to stay for the next decade. (Unless they start screwing up their deal in which case I'm sure simple would still love to have me -- only thing I don't like about them is I prefer having a mastercard debit card rather than visa)

Hope any of this info is useful to somebody.