Now I only know, from fastest time controls, which exist on chess sites, 1+0 and 0+1. But is somewhere a possibility to play half-minute games? Thank you.
If you play on a tournament in http://lichess.org, you have the option to go into Berserk mode. This effectively halves your time (NOTE: not your opponent's), for the possibility of winning more points. If you're playing a 1 minute tournament, then berserking every game will allow you to effectively be playing 30 second games.
That being said, you won't learn much by playing 30 second games because the game will almost always come down to who's faster with the mouse as opposed to who's a faster thinker... but I'm assuming you already knew that.
EDIT: Since I originally wrote this, Lichess has implemented a stand-a-lone 30 second tournament mode. That is, both players are automatically forced to use 30 seconds for the whole game without having to berserk.
The website http://chesstempo.com allows 10 seconds bullet games as the shortest possible.
Chess.com now allows games down to 10 seconds and no increment. However, it's hard to find anyone playing faster times than the normal 60 second bullet. I just played a 10 second game after seeking for about two minutes.
I pulled this announcement off Facebook just now:
I tested how low the time control is allowed to be set. I found that the system seeks for a ten second game, but if you input 9 seconds as the time control, it will reject the input.