When facing a 1BB min-raise, I typically view it like a check. If I would have checked, I’ll call; if I would have bet, I’ll raise. There are rare instances where this is the appropriate bet size, like a J22 board and the pot is just 4-5BB.
Min-raises to a different size are a different story. Usually it’s a chance to play perfectly with a polarized range - fold off my bluffs and call with my strong value.
If everyone limped to the flop, I can understand a min bet afterward. I wouldn’t do it as I think it’s weak, but I’m aware others are slowplaying or as Bill with all the 8s says, controlling the pot (usually when they’re on a draw). I also agree with Wannabe as to how to react, calling if I would have checked, raising if I would have led out, folding if I’m more than one card away from a made hand.
The min bet that gives me aggro is the one after the pot has been raised. Let’s say the blinds are 20/40, a preflop raiser has made it 120, three people have called and the small blind has folded. There is 500 in the pot. A min bet of 40 chips is insulting in that instance. It’s not even ten percent of the pot.
I know some players will do this as a variant on check raising. They min bet. Someone raises to say half the pot and they call. They do it again on the turn and the river if the opponent doesn’t catch on. In other words, they let you supply the rope to hang yourself, then they show a strong top pair/top kicker, two pair, or a set.
Paying attention to the hands you’re not playing to be aware of this tactic, as well as sizing your bets to keep this type of fish from controlling the pot are important. Assuming you’ve hit the flop yourself, don’t min raise them back, but neither should you shove or push too hard. On the flop and turn, I recommend something between a quarter and a third of the pot (125-50 in the scenario I laid out), then just call on the river so that if their hand proves the better, you haven’t crippled yourself trying to intimidate them.
If they do this when you’re on a draw, call to the river as the price for each card is good value, but don’t bluff this player if you miss.