I stand on mine. It's an emotion kayaks renegade XT. It's really stable cause it's wider than most kayaks but if you want to stand I would recommend a wider one. Also look for one with an easy accessible dry hatch while on board, trust me. Also, you'll want to consider weight cause I'm sure you'll be doing a lot of your fishing by yourself so you want something you can manage and launch on your own.
Don't skimp on your paddle either. You'll want a nice light weight paddle so you don't get fatigued. Lastly if you get the chance, sit in the kayak before you buy it. You might find it does or doesn't have acceptable back support for you. All kayaks feel different and you want to be sure you'll be comfortable all day.
Good luck.