Q:

Can the Runner/Corporation trash their own installed cards? 

This mostly came up last game with a player that had Wyldside. 

When your turn begins, draw 2 cards and lose [Click]. 

He was using it to good effect up until a few turns in, when he had too many cards and not enough money to do anything with them.  The question becomes, can he voluntarily trash his own installed cards without a prerequisite action? 
I know that the Corporation can trash their own installed agendas/assets in servers they control by simply installing something over what they want to trash, and Runners can trash programs by installing a program that would be above their storage limit.   But for Wyldside, there's no way to trash it unless you have Aesop's Pawnshop. 

A:

Unless a card states otherwise, players may only trash cards in the following circumstances:
In the rulebook (PDF) page 13:

When installing a card in a server, the Corporation can first trash any cards already installed in that server.  Trashed cards go to Archives faceup if they are rezzed, and facedown if they are unrezzed. 

This means that under normal circumstances, Corps can trash ICE (rezzed or unrezzed), assets/upgrades (rezzed or unrezzed) or agendas when installing any ICE/asset/agenda/upgrade in that same server.  They might do this to install a valued card in a well-defended server rather than creating a new server, or to reduce the install cost of ICE.  They MUST trash assets/agendas in servers that already contain assets/agendas. 
Corps cannot trash ICE/assets/agendas/upgrades at any other time or in a different server. 
On page 15:

The Runner can choose to trash any number of his installed programs at the beginning of an install program action. 

Under normal circumstances, Runners can trash programs when installing a program.  In fact, they MUST trash programs if their MU limit is exceeded in this instance. 
Runners cannot trash hardware/resources (including Wyldside!)  or programs at times other than the above. 
But, a card's text can over-rule this.  For instance, Wyldside cannot be voluntarily trashed of its own accord, but, as you correctly mention, a card like Aesop's Pawnshop can trash it (and for a benefit) because the card explicitly tells you to trash something:

When your turn begins, you may trash another of your installed cards to gain 3 [Credits].