Some puzzles have particular design features that you might be looking for:
Irregular edge - extra hard
Straight edge - a bit easier
"Normal" shaped puzzle pieces but with whimsy pieces - a bit easier
Double-sided - extra hard
Escher inspired tesellations of pieces - extra hard
Drop-outs - extra fun
Bigger pieces - a bit easier
Super-fancy piece shapes - extra fun
Split tendril connectors - really hard!
Nestled spiral pieces - extra fun
Smoke-ring pieces - bit easier and extra fun
Complex connectors - extra fun
Multi-piece whimsies - extra fun
Puzzles where pieces can fit together wrong:
- Non-unique connectors - extra hard
- Mini hexagon pieces - extra hard
- Connectorless - extra hard