For Selfies:
I have a Lumix G3, and in the past used a Canon Powershot pro 1. Both of these are semi-compact and importantly have flip out screens.
You don't need a tripod if you carry a small bean-bag around with you.
1. Keep fish in water in the net.
2. Place camera on bean bag and (with screen flipped) frame the shot.
3. When you're ready hit the shutter (on 10sec delay).
4. When camera starts to beep/flash fast, gently lift fish for it's portrait.
5. Unhook and return fish.
For non-selfies
1. Keep fish in water in the net.
2. Fine nice shallow spot (ideally say 3 or 4" deep) where fish can stay wet but you get a good view of it.
3. Photograph fish flat on net (still in the water!)
4. Unhook and return fish.
Just a personal thing here, but I do dislike all these one-handed selfies with the fish being gripped between thumb and fingers. Just cradle the fish in your fingers! If it jumps, tough. (or if you're desperate you could delay unhooking until the photos done - allowing a bit of re-wrangling if it does slip from your grip.) Again - just a personal thing. No offence
W.