Type-A: Shimano and Look cleats have little rubber grippers that provide stability when walking in cycling shoes. Once those are gone, the plastic that engages with the pedal starts to go. This is a good time to swap for a new set of cleats.
Procrastinator: Abuse a cleat enough and it will become loose and wobbly when engaged with the pedal. Speedplay cleats eventually wear to the point that the mounting screws file down and become fixed to the sole. Don’t let it get that bad.