In the "snaps" tab, I typically have one or the other selected. (grid or vertex snapping) Using grid snapping in any kind of perspective view is usually not usable (unless creating a primitive on the grid.) Just in case this helps you find your solution.
Checking/unchecking axis constraints does nothing, apart from what you'd expect it to do, which is enabling/disabling the axis constraints. The dropdown menu is set to standard. This is what my Grid and Snap Settings window looks like: Unfortunately I already knew all that, but thanks anyway.
Thank you all for your replies. My snap settings are set to snap to vertices, and I also tried snapping to the grid. As I said before, snapping works, but only when creating objects. It doesn't work when moving or editing objects, though, and that is my problem. I always use the gizmo handles for moving. Grabbing the…