There have been several questions on meta recently that bring up a similar topic, but I would like to formulate the question explicitly here:
Which puzzles that use a chess board are on-topic here?
To make the answer easier, I would like to divide such puzzles into several groups:
Puzzles that use a chess board only, like covering chess board problems, for example The Mutilated Chessboard. Or cutting chess board problems.
Puzzles that use a chess board and modified chess pieces. That means pieces which moves differently from usual 6 types of chess pieces. For example, The Bunny's Tour, How Many Speedy Knights on a Chess Board?.
Puzzles that use a chess board and chess pieces (including the rules of their movements), but do not care about the game itself as a set of moves leading to a win or loss. For example, How many queens can be on a chessboard without attacking each other?, How many squares can 4 queens control?, How many chess pieces does it take to "cover" all spaces on a chessboard?, Most moveable chess position, Optimal strategy against a random opponent, Find the shortest chess game with 18 queens?.
Puzzles that use a chess board and chess pieces and full game rules, but ask questions that are not really useful for chess players, but interesting from a rules investigation point of view. For example, What is the position in chess with the longest string of moves before mate?.
Retrograde analysis chess puzzles. These can help you to learn to play chess on some level (unlike the previous category), but they ask for questions, that never arise during real chess games because there you know all previous moves. Also, they usually provide chess positions that would never appear in a real game. For example, Is there a retrograde analysis puzzle like "Point on the incorrectly drawn chess piece"?, Find the color of the chess pieces
Traditional chess puzzles, like "White wins in 2 moves," "White starts and wins."
Also, we have chess tag already, so another question is: Which type of puzzles should be tagged with the chess tag and which should not? Maybe some less-chess-related puzzles should be tagged differently, like the puzzles in category 1.