It is evident that bullet chess has a different set of strategies from regular chess. It follows, therefore, that certain openings will be better or worse relative to their normal chess counterparts. Hikaru Nakamura covers some examples in his book "One Minute to Mate", but the coverage is very light. What are some good bullet chess openings that are particularly strong due to the nature of bullet chess?
The best bullet players can and do play standard openings. The openings are theory because they are the best moves.
If you want to play something different, there are a few justifications. If you play certain moves all the time, you can often make the moves faster and gain time on the clock. Similarly, if you play something off-beat, you will most likely know some plans from the position and often have a better understanding than your opponent.
With that lead in, one of the simplest openings to play is to fianchetto both bishops, play
The main plans of this opening are to develop all of your minor pieces and then decide to play either
Personally, I have always had better success in online chess (especially bullet) playing standard openings. Play openings that you are comfortable with, and you will be able to take advantage of your whole sum of chess knowledge, as opposed to just utilizing bullet "tricks".
While it is beyond the scope of this question, one of the best ways to improve at bullet is to work on playing better and faster when both players have 10 seconds. Most games are won or lost in this time frame.
Openings you are more familiar with than your opponent, and in which it makes a difference¹. If you want to take advantage of it somehow, and not just “play better”.
Therefore I'd propose to go for gambits and very open positions, in an agressive style². Especially not well-known ones³. In these, you being familiar with the ins and outs of them may well give you a big advantage if your opponent isn't. On the contrary, in a closed game, tiny mistakes can be a lot harder to punish, especially in this time frame (i.e. you don't want to have so safe-but-small an advantage the first slip you do in endgame will ruin it).
My attempt at name-dropping, I'm sure many can do better :
And for black :
If you're ready to learn many more, you can delay them to later moves in most played lines, but that grows the repertory you have to master. You might want to have a look there.
¹ I'd argue Andrew's suggestion of fianchettos is way too passive for that, as your opponent has virtually no chance of doing a mistake against that. (Also I've played this system for a long time, though not restraining myself to blitz or bullet.)
In my opinion, one must play an opening that one plays most often, as in bullet chess there is very little time to think. If you are receptive to go and learn a new opening, then I recommend Stonewall as it makes the game a very closed game while retaining a strong hold of the center. With black pieces, French Defence and Budapest Gambit serve as good openings.