Let's start with a classic: the missing granny! This is an issue that has been known for a long time and for which Bioware customer service generally seems to recommend running the repair utility on the launcher. Fortunately for me, I never actually encountered this myself until late last year, and even then it was just this one class mission during which Mako's gear was suddenly missing a texture, and it quickly seemed to fix itself without me doing anything. The jokes about missing elderly people never get old though.
One of the patch notes for patch 3.0.2 stated: "Players beginning the Mission “Torch's Flame” can no longer see duplicates of Lana and Theron." To which I'll say: I'll have your two Therons and raise you two Jakarros! (This was one of the later missions too.)
Once I was questing on Republic Balmorra with my pet tank and he said: "Look, a floating tree!" Huh. Well, that's one I hadn't seen before. And I'm pretty sure that it's not usually like that.
And finally, another visual glitch that has been around for ages: the lost assault cannon.
Every so often my character will appear to "lose" her assault cannon, which means that my healing beam appears to originate wherever she "lost" it instead of from her actual position. I used to find this kind of irritating, but nowadays I'm mostly amused by the light show, especially knowing that it will always fix itself pretty quickly.