As of Jan 2018, the FIDE rules are a lot clearer than they used to be. I am particularly happy with 3.10 (which arrived in 2014 I think), which defines legal moves, illegal moves and illegal positions. (Curiously, legal positions are not defined, but I can we can safely induce what they must be from the other three definitions.)
I would like a term for what a legal move becomes when it is unplayable because the game has been terminated (through dead position, repetition, whatever... basically any cause of termination apart from mate & pat, when by definition no legal moves remain). But that's not really an ambiguity, just a lack of standard terminology.
One area of historical ambiguity has been largely cleared up: and that's the impact of castling and en passant rights on draw by repetition. Residual confusion is perhaps because the basis is different between castling and en passant. Castling rights depend purely on whether a king or rook has been moved or captured. There is no investigation of the possibility of actually castling. On the other hand, en passant status depends on whether the capture can actually be executed, or whether it is prevented by some direct or indirect pin or check. This distinction (which biologists might term "genotypic" versus "phenotypic") is completely rational, but I think that some people expect castling and en passant to be handled in the same way.
A final corner-case cobweb in this area remains: suppose king and rook have not moved, and then the rook is captured. Does the relevant castling right remain? 126.96.36.199 says "The castling rights are lost only after the king or rook is moved." This could actually have an impact.
There is the famous issue of checkmate on the 50.0th non-capturing, non-pawn move. To me it is clear that the player has the choice over the board to draw if they want to, just as a player is allowed to resign in a position which is clearly not lost, or offer a draw in a clearly won position. I don't see why an arbiter should feel the need to interfere in the first case, but not in the other two.
Two other lacunae, other posters have competently detailed, but I will list for completeness:
(1) Both players resign (race condition).
(2) No-one has the move at game start.
In summary, as a problemist, the main change I would still like to see in the FIDE Laws is the completion of the partition into (1) rules of play, which are defined mathematically, requiring no arbiter (2) over the board conduct, about embedding the mathematical object that is a game of chess into the real world, and where the arbiter's judgement rightly holds sway.