I'm someone who often finds herself changing avatars/sigs a lot, sometimes they only lasted one day but I'm slowing things down recently.
As for my fave avatars/sigs that I've used here, nothing can beat my iconic Oshawott themes. It's my fave Pokemon and I made so many avatars for it out of various anime pics and fanart, so it's pretty much defined as a mascot for me.
Some other fave themes I've had in the past included Froakie (my second fave Water starter ever), Marill (the first theme I used when the current supporter system came in, as part of a collab), Marlon (the Water Gym Leader from B2/W2), The Blue Umbrella (it's a Pixar short film), EVE from WALL-E (one of my all-time fave Disney/Pixar feature films), May (her OR/AS design, which I think is the best-looking May ever), some Homestuck characters (especially Terezi because she's my patron troll, given I'm a Libra) and a few music-based ones I've had, including Zedd, Empire Of The Sun, The Black Keys, Lorde, alt-J and Foster The People (which happens to be the one I currently have up). Oh, and also that recent avatar I had of Mordecai from Regular Show because I think it looks nice with my name color.