Basically all of them!
Chess engines really only use brute-force when:
- told to
- are analysing positions (problem solving)
- Looking for a checkmate (problem solving, not when playing against, like "find the mate in N" style problems)
Otherwise they have a "selective search", this will consider all the possible moves for a given board layout, but only explore a handful of them. An engine may switch to brute-force though if it rates two moves very similarly (more than one strong move) or if it cannot find a move it likes (no strong moves).
They also tend to brute-force as a last line of defense, if you've seen a checkmate chances are it can see it coming and it will want to try really hard to draw, and cannot find a way out (the "Horizon effect" is a problem with engines, suppose it is going to loose it's queen, and it's been capped to only go 4 plays deep; if it can trade pawns and postpone that loss of the queen for 4 moves it will think it has saved the queen, in the process it'll loose at least 1 pawn (as the next move brings the horizon from before closer) and the weight it places on saving the queen may mean it sacrifices some defense, for nothing if the death goes over the horizon).
It will also brute-force when the selective search isn't very useful. This is why engines take longer when they have like 3 pieces left. They have to brute-force because the selection algorithm cannot rate a move. The selection algorithm is great during the midgame because it can be like "Oohh, doing this with the pawn blocks his [whatever] and backs up my [whatever] and [whatever] which I have a lesser number defending than attacking" - for instance.
If you have a king in the middle of the board there are 8 moves, the selective search will be like "None of these do anything useful, I can't tell".
You can think of the selective search as having two parts, it is tactical in the sense of it'll try and spot tactical moves, it will ignore the weight of the pieces involved usually because a queen not a part of any strategy is not worth more than a pawn vital to it. It is also strategic in that it'll explore moves that bolster a defense, and open up later to potential attacks.
The engine then does the same from your point of view, and back and forth and back and forth.
Something called a transposition table is a big list of things it has thought about, that way if it ends up considering something it has already done, it knows and doesn't have to re-evaluate it.
UNLESS (selective :)) it gets there a different way, or wants to explore further. Suppose for example it discovers that your ... rook is essential to an impending attack, the engine may re-evaluate a line when it discovers this. The previous weight it put on that rook (eg 5 points, how important it is to you) might be an under-estimate.
The selective search can also backtrack, like say its considering a bishop moving right into enemy territory, to the move selector it is not important that it can be taken easily. Say it discovers that strategically that is a superb move! It may then backtrack to try and find a way to protect that square to get that bishop there. Suppose it involves a pawn to do so.
The brute-force method would consider the line involving that pawn move, and (by brute force) the bishop move too, and the same stuff that rates the board position (the selective search itself) will say "this is good" so the board rates that variation highly, both find it.
It's very hard to rate a position using the brute-force method, this is why the selective search works so well.
The brute-force from the starting position might find that famous mate-in-4 that involves a queen f7 covered by a bishop, and if it were to rate that highly (I'VE FOUND A CHECKMATE! JOB DONE! PLAY!) it'd be wrong because black will obviously counter. The selective search rates a position (for further evaluation) because it is appears to be good. This means when it is considering your response it can decide what would be good for you....
So the stuff that the selective search uses to rate things is used by the brute-force one anyway because "found a checkmate involving this move" is not enough to say that move is good.
Hence What are the first moves chosen (White), by brute force chess engines?