I bought the Legendary Edition of Skyrim a while back and decided to start a new character. The first problem I ran into was that the carts in the opening liked to wobble on the road, almost toppling over. Then, after exiting the character creation screen, the headsman was off his mark, causing him to not execute the Stormcloak (or myself when I used the "kill" command on him.) after restarting, he was on his mark and the game normally (though the horse somehow got hurt before the title had a chance to fade in.) until I entered Alvor's house. some peice of furnishing was constantly rattling like it was stuck on something. Then when I noticed many of the animals kept teleporting up a few feet at random intervals, sometimes killing themselves this way. The waters liked to flicker at times, and in Moss Mother Cavern, it would act like they were swimming at random ticks, making navigating the cave a chore. Also, in Fort Greymoor's dungeon, opening cell door would cause loose object several feet away to fly across the room. Opening chests would have a simalur effect on nearby items, too. Breezehome and the Bannered Mare were two other places with the rattling problem, with some things being thrown with the slightest touch. Then there were more scripting issues. Mirmulnir's body wouldn't burn up after death (maybe I wasn't the Dragonborn that time, heh.) and after another new char., Alvor's wife would get stuck in the stairs, thus preventing early progress in the main questline.
I tried capturing a few of these glitches with FRAPS, but in the 30 seconds I record, the game acts normally. I ended up filming the bugs with a camera at the screen instead. Crude but it worked. I have the Unoffical patch installed, but turning it off didn't seem to fix anything.
Specs: i5 3350p 3.10 GHz quad core
EVGA GTX 650Ti
8GB Ram