Here are some types of irregular grids common in WPC-style puzzles.
In some puzzles, the geometry used in it is conventional square one, but the cells are partitioned differently.
Examples of originally hexagonal puzzles include Boomerangs.
Row/Column: Translated into 3 directions. For example, this is hexagonal Easy as AB from PGP IB.
Straight/Right Turn: Some puzzle forbids 60° turns at all. Some puzzle introduces a different notation for loop being straight, making 60° turn, making 120° turn.
Straight/Right Turn: Loop makes 120° turn in every cells.
Row/Column: Undefined, since there are no straight direction.
Straight/Right Turn: Translated into: "when edges that the loop crosses when entering and leaving the cell are adjacent, the loop "makes a turn" in the cell." This is Cairo Pentagonal Masyu from WPC 2019 IB.
Cylindrical, Toroidal, Möbius strip
In a cylindrical grid, the leftmost and the rightmost rows are considered adjacent.
In a Möbius strip grid, the topmost cell of the leftmost row are connected to the bottom cell of the rightmost row, et cetera.