a3/h3 are frequently used to prevent the bishops from pinning the knights.
I always have a hard time deciding whether to play them or not. Playing them loses moves and tempo while not playing can also sometimes become annoying (for example, if the knight is pinned after the bishop has already moved OR worse, if the bishop cannot come back to remove the pin).
Can the experts here help the rest of us with some thumb rules and/or examples?
Let's stick to one color, white here, and everything we'll say will generally hold true for black as well.
Rule of thumb: On the one hand, you generally don't want to move your king-side pawns (such as h3) when you've castled short, unless you have to, and we will expand on what "have to" entails here. The immediate exception to that is when you decide to fianchetto your king-side bishop, e.g., in the Catalan where then you play g3. On the other hand, your queen-side pawns (assuming you've castled short) are to be used more actively to either gain space or prevent an spatial expansion by your opponent. (the rule is naturally to be taken with caution, chess rarely lends itself to these kinds of generalizations)
The more common cases are when a bishop is pinning your knight on g4 and h3 is an immediate way to try and challenge the bishop, asking it where it wants to go. But there are cases where g4 is not being exploited for a pin but rather as an intermediate square to re-route a knight (to e5 for instance). In either case, if the pin is problematic to deal with, which may be so if you've already made a commitment with your light square bishop and it cannot return to e2, or if the knight re-routing would be an improvement for your opponent, then you really want to play h3 in a prophylactic way to prevent all of that. Here's an example in the Benko gambit:
[Title "Benko gambit declined - preventive example"]
[fen ""]
1.d4 Nf6 2.c4 c5 3.d5 b5 4.Nf3 g6 5.Qc2 Bg7 6.e4 d6 7.h3 {Stopping any future knight re-routes to e5 via g4, and specially securing e3 for the c1 bishop. Bg4 is altogether less of a concern here in comparison}
And below you find two prophylactic examples of a3 and h3 where they are purely strategic and intentional:
[title "a) Nimzo-Indian b)London system - (deliberate examples)"]
[fen ""]
1.d4 Nf6 (1...d5 2.Nf3 Nf6 3.Bf4 e6 4.e3 Be7 5.Nbd2 O-O 6.h3 {this is largely to secure the h2 square so that we can preserve our dark square bishop for instance against Nh5.}) 2.c4 e6 3.Nc3 Bb4 4.Qc2 O-O 5.a3 {white by choice, not forced, plays a3 to resolve the pressure on c3 in order to finish development more deliberately, with choices such as Bg5 becoming easier to make} Bxc3+ 6.Qxc3 b6 7.Bg5
Then there are reactive cases where tactically one may not have a choice but to challenge a piece on g4 or b4 respectively, in which case although there's no doubt your opponent would be forcing your hand into playing h3 (a3), but you will have to include that move and accept a small set-back. Extreme cases are when a mate is threatened, e.g., a queen on h5 and knight on g4.
Why moving the castled-side pawns would generally be a set-back?
Main reason for that is two-fold: First, as obvious as it may sound, you'd be permanently committing to a pawn structure, second and tightly related to the first, you'd be creating targets for your opponent. The latter may be, to name a few, in the form of weakened dark squares, for instance if there's a dark-square battery on the c7-h2 diagonal that you'd want to preferably deal with by playing g3, but having previously played h3 may make that difficult and g3 would expose h3 and g3 itself to potential sacrifices. Another type of target would be, a pawn push on the king-side by your opponent, challenging h3 with g4 attempting to undermine your king-side pawns and open a file towards your king.
The takeaway here can be generalized: pawn moves in chess are always a commitment, the later you have to commit to them, the more pawn structures you can opt for, and conversely, the sooner you commit to one, be it willingly or forced by your opponent, the earlier you create targets for your opponent. Targets are not always weak pieces/pawns (such as back pawns, or isolated ones), but they can also be strategic targets that can be exploited in your position, prime example being weakened-black/white-squares, or more abstractly, your pawn-breaks in the position can become limited, and the few remaining ones can more easily be prevented/delayed by your opponent, which ultimately means you will have a hard time undermining your opponent's structure and creating active play. This is closely related to space control in chess, which you can read more about here. With all of this in mind, now you can see for example why the English opening is generally is hard to face: it is extremely versatile in pawn structures it can lead to, consequently, the opponent has a harder time to resolve the dilemma of which plan to opt for! Instead, e4 and d4 openings are more committal by definition (as we start by moving a central pawn already), so we can more reliably establish systems against them
Now let's take an extreme example where our rule of thumb is being perfectly illustrated:
[title "Michael Adams vs Garry Kasparov, Linares 2005"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 a6 6.Be3 e6 7.Be2 Qc7 8.Qd2 b5 9.a3 Bb7 10.f3 Nc6 11.O-O-O b4 12.axb4 Nxb4 13.g4 Be7 14.g5 Nd7 15.h4 Nc5 16.Kb1 Rb8 17.h5 O-O 18.g6 Bf6 19.Rdg1 Ba8 20.Bg5 Be5 21.gxh7+ Kxh7 22.Nb3 Nxc2 23.Nxc5 Na3+ 24.Ka2 Qxc5 25.Na4 Nc2 26.Kb1 Qa3 0-1
Notice how Kasparov pushes with a6-b5-b4 on the queen-side in order to launch his attack on white's long castled king and yet never moves any of his king-side pawns until the end as to avoid giving white any entry-points or targets. Notice how Adams had to play earlier a3 in order to prevent b4 as it would force his c3 knight to be mis-placed in turn weakening the e4 support. And having committed to a3 is precisely the target Kasparov used with b5-b4 to open the b-file, which he then exploited beautifully. This is a cherry picked example, but it's very appropriate here because all of our previous points culminate in one game! More generally, Sicilian players are accustomed to this type of pawn structures (i.e., knowing not to commit their king-side pawns and just push as fast as possible on the queen-side).
NEVER say never. Rules of thumb are just that. White very commonly plays h3 at an early stage in many variations of the Ruy Lopez, even before Black has committed his K. What often makes this safe is that Black may have already disarranged his own Q-side pawns by ..a6 and ..b5, so he is unlikely to castle over there.
I remember learning this rule of thumb for White in the Lopez. Do not play d4 if Black can answer with ..Bg4, because then your center will be under attack. But if Black plays ..Bg4 early, go with d3, because then your center is not under attack and you can manoeuvre ..Nd2-f1-e3 with tempo against the misplaced (?) Bishop if you have not played h3.
The answer to your question is that h3 (or any other move) is good if it forms part of an integrated plan. Rules of thumb like the ones quoted by Glorfindel or Phonon or myself are useful if they start your planning off in a good direction, but the thing about plans is that you must be prepared to change your mind (and perhaps change it back again). To reinforce a point made by Phonon, I have always admired (but not always remembered) the advice of James Mason,"You have 48 pawn moves. Spend each one as if it was the last dollar between you and starvation".
Reasons not to play h3 in the Lopez; 1. It encourages a charge by the Black g-pawn (but is that anything more than just a bit scary? Does Black have backup?) 2. It weakens f4 to occupation by an enemy Knight because g3 will now hang the h-pawn (Is that Knight currently anywhere near f4?) 3. It may lose time for executing a desirable manoeuvre elsewhere (but does that matter if the manoeuvre is merely delayed and cannot actually be prevented?)
(Who said that h3 was "provincial timidity occasioned by metropolitan excitement?", Staunton?)
Reasons for playing h3 in the Lopez; 1.To prevent Bg4 (if this was really a threat) 2. To prevent Ng4 and secure stability for a B on e3, 3.To remove back-rank anxieties 4. To keep more pieces on the board (if Black is already cramped. 5. to prepare g4 (if you are the one ready with a pawn-storm.) 6. As a waiting move (If Black is trying not to commit to some decision but you want them to reveal it) 7. To drive off an attacking Knight before reinforcements arrive.
There can be many more and often they are in conflict. What you must do is try take all of these into account (and combine them with decisions about your c-pawn, for example) until you have a harmonious plan (Much of this will eventually become instinctive, as part of your "chess vision", the pieces will look neat and organized.) Then try to remain consistent with that plan (accentuate the positive, eliminate the negative) until circumstances change; be very alert for that moment. If your plan does not work, recognize that you have been outplayed by a better planner (unless you just missed a Knight fork). The focus of a postmortem should not be "Was this a good move?" but "was this a good plan?" Rules of thumb are invaluable as reminders in the early stage of planning. Ask, What is the usual situation? How might this be different?
Never do it in front of your castled king (or on the wing where you want to castle), if the opponent hasn't castled yet or castled on the other wing. Doing so weakens your king's fortress and is an open invitation for a storm by the b- or g-pawn.
Generally, in open games (after 1. e4 e5), and especially with Black, the loss of tempo involved in a (preemptive) a3/h3/a6/h6 will be heavier than in closed games. Actually, I have a hard time thinking of main line openings where this move is played to prevent the bishop from coming to b4/g4/b5/g5. That alone should be a reason not to play it.
Playing it after the opponent moved the bishop (e.g. 3... a6 in the Ruy Lopez, 4. a3 in the Sämisch variation of the Nimzo-Indian) is usually less of a problem, because it forces the bishop to retreat or be exchanged with the knight, giving you the bishop pair.
