Connect Wall - Grid Puzzles Edition!

_{This is an entry for Fortnightly Topic Challenge #42: Wordless Connecting Walls.}

_{Huge thanks to bobble and Deusovi for playtesting! ><
And hey, this one may also mark my 100-day puzzle here in PSE, aye-aye the golden Socratic badge! XD}

The $16$ grid puzzles above should be divided into $4$ categories of $4$, based on some shared properties.
For the final answer, please find one or two grid puzzles that satisfy all $4$ categories!