I'm not one of the mods, but looking at all emulated runs on here I noticed they all use FCEUX.
I don't think that's necessarily a required emulator, though. There's a bunch of NES emulators and on my previous laptop FCEUX just happened to run terrible, so I played my own games with Nestopia. Nowadays I'm playing games with Mesen.
Here's a few things to think about in an emulator:
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Does the emulator run well on your computer while recording(FCEUX/Mesen/Others ran awful once I started recording on my previous device).
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Does it show input? Not necessary, but usually highly appreciated. Nestopia didn't have this, so I switched to Mesen now.
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Being able to show a frame count. This one's usually only for popular games and when you're going for WRs.
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Is the emu somewhat accurate? FCEUX is pretty low on this, but it still runs fine for most people, so obviously this isn't a requirement. As long as it plays correctly(no emu specific things that a console player can't do).
No emulator will be exactly like playing it on console, but I think with the above you have pretty good coverage with what could be considered a speedrun friendly emulator. Assuming you don't use L+R, turbo, etc. any emulator is likely fine, but if it doesn't run well, I suggest you pick another one instead of suffer through massive lag, skipping, etc. :P
Every emulator that is approved by SRL applies to speedrun.com aswell for the most part, unless the specific communities have different rules.
Personally, I've always used FCEUX. I fully recommend it!