If I have ten minutes’ spare time and want to use it to improve my chess, is my time better spent playing one blitz game or five bullets? (I’m taking blitz to be 5min each and bullet to be 1min each) (I’m ~1300 - 1400 rated)
What improves your chess is the time you spend thinking on chess and solving chess problems (either during a game or when doing exercices).
As a consequence, if your concern is to make progress, a 5' blitz is better than five 1' bullets.
What you will play will look more like a real chess game; at 1300-1400 I suppose you seldom drop pieces in blitz, but often in bullet. Also, maybe this will be the occasion to practice an opening you would use again, actually thinking at one or two points in the game on "What do I do next ? What are my plans ? What is my opponent intending ?". At least at two or three times in the middlegame you will slow down and calculate a few lines. Calculating is useful training. Maybe you will still have thirty seconds or one minute to invest for an important decision in the rook endgame; in bullet the endgame is never reached, or butchered until a player gets flagged.
The fast, repetitive, mechanical play, which is almost mandatory in bullet because of time pressure (also in blitz, actually, but less so) is actually rather hampering your progress, because you keep making bad moves and teach yourself to play automatically, not calculating. Also playing for tricks and cheapos (if he doesn't see this I will fork his queen...) is the wrong mindset for when you'll play real chess.
Anyway, don't expect too much progress from a 10' practice. The blitz might or might not help a tiny little bit, but you will need to invest more time and various training (play, analysis, exercices, books/videos on openings/middlegame/endgame, etc.) if you want to achieve significative progress. The bullets, if anything, are counterproductive.
For the 1300-1400 level, none are useful if you want to improve. More often than not, blitz and bullet (especially the latter) just develop bad habits, such as moving too quickly or having little patience for in-depth analysis. If you want to work on improving in 10 minutes, I'd suggest doing a few tactics, solving a study, or playing through a master game.
Blitz is only useful for some players after crossing the 2000-2200 barrier. The reason for this is that blitz can help develop more intuitive and quick thinking. Most importantly, an expert/master knows that blitz chess is different from long games, and so they can ideally not let their blitz habits interfere with their long game habits.
Bullet on the other hand isn't useful for any level of player. It's only purpose is for having fun. You can't develop any good habits with only 1 minute to think for an entire game. It's likely most of the calculations you do will be very flawed and inaccurate, and then this can "leak over" on to your long games.
Neither of your alternatives is much use for learning anything except bad habits and sloppy thinking.
Bullet chess is completely useless for improvement. Magnus Carlsen might pay better chess than you or I with only 30 seconds on his clock, but he is one of the strongest players of all time. With 1 minute or less for the whole game most players produce pure rubbish.
Five minute chess is not much better. You have to rely mostly on intuition, but unless you are already a very strong player you won't have very good intuition. Yet there isn't time to think through alternative placements of the pieces, nor to calculate deeply. So once again less than masterly players just play junk.
If your only aim is to have fun then play as fast as you like. If you are serious about improving you have to play slow chess. Not only will speed chess not allow you to develop positive knowledge, calculating skills and good intuition but also it will almost certainly get you into bad habits, such as playing the most obvious moves all the time and not making a blunder check after every move.
I reckon that for anyone under 2050 strength (and that includes me) the absolute minimum clock time for a useful learning game is 15 minutes each. But 30 minutes is better and classical time limits (1 hour 40 minutes or 2 hours for 40 - with or without the Fischer increment) are better still.
The accepted answer is completely fine but adding my two cents.
Play a 3+3 games. I think playing games with increment are better for improvement, you could Blitz out moves but also use some think to think and analyze when needed.
Use the time left to quickly go over your Mistakes.