Just to be a little more explicit with that... the rules say that ZSNES and Snes9x 1.43 are banned. Almost any other emulator will be legal, but the most common are higan, Snes9x 1.53 or greater, and Retroarch.

The top runners tend to play on original console to reduce input lag, but a ton of us in the community play on emulator, it's certainly not uncommon.

With Retroarch, is there a preferred core to use?

The Snes9x core (with no date attached) is very common, as seen in the ALttP community's recommendations: https://alttp-wiki.net/index.php/RetroArch_Settings

That said, the core of any other allowed emulator is also fine, such as bsnes and higan.

While we're on the topic, why isn't ZSNES or older versions of SNES9x allowed?

Mostly due to improper emulation of processing lag. Certain areas of the game cause heavy lag on console, such as Kraid rising from the ground, or setting off a Power Bomb in most of Lower Norfair. Inaccurate emulators might not have the slowdown that real hardware does, giving them an advantage here.

It's also worth noting that zsnes has a serious security flaw that can open your computer to be compromised. For these reasons and others, the Link to the Past Randomizer actually checks if you're running on zsnes, and fails to run if you are: https://www.reddit.com/r/alttpr/comments/8gfgr9/please_dont_use_zsnes/

As a result, our recommendation is bsnes if your computer can run it, and Snes9x 1.5 or greater otherwise.