They nearly all have a self-timer setting so you can start the clock, race round the front, pick up your fish, smile, wait, smile a bit more, and then put the fish down and be photographed just as you walk back to the camera to see what's going on.
This happens because ten seconds (the usual delay) is quite a long time. On some you can extend the delay time.
All modern compact cameras have auto everything, making them pretty much ideal for this kind of thing.
But if you look around, you may find models which have a screen which can be faced forwards, so you can actually see yourself being photographed. Not sure which modern ones have this, but my old Nikon Coolpix 995 can be swivelled so you can see what you are taking a picture of. I think one of the poncey Fujis has a screen you can flip around. Could be wrong.
Without this, there's always the chance of tailless fish and Max Wall haircuts
