NES

• PocketNES: Best one for GBA. Fast, stable, with multiple zoom options.
• InfoNES: Old yet decent emulator.
• Famicom Advance: Slower than PocketNES. FC/NES emulation.
• HCVA: FC/NES emulator with support for lots of mappers.

SNES

Not really ideal for the obvious reason GBA has two buttons less than the SNES.

• SNES Advance: Formerly known as PocketSNES. Different versions have different game-specific fixes breaking stuff elsewhere. This emulator is very inaccurate and has lots of issues, it hasn't been updated for a long time, so it's not recommended to use.
• Pocket SNES: Low Compatibility. Download

Sega Genesis

• No known emulator for Genesis yet.

 Sega Master System / Game Gear

• DrSMS: GG/MS. Excellent speed and compatibility.
• SMS Advance: SG-1000/GG/MS. Good. No FM Chip, EEPROM emulation.

PC Engine

• PCEAdvance: Very decent emulator.

Handheld

Game Boy Color

If you don't want to use a GB Bridge, emulators exist as well:

• GBonGBA
• Goomba Color: Sound, SGB borders, savestates. (Updated Link)
• PocketGB: Low compatibility, high speed.

Neo Geo Pocket

• NGP Advance: incomplete but good.

WonderSwan:

• SwanAdvance: Very early, low compatibility and speed.

Arcade:

• Greenboy Attack: Green Beret, Rush'n Attack, Mr. Goemon.
• RGBArcade: Donkey Kong 1,3 and Jr, Gorkans, Pacman and Ms.Pacman, Phoenix and Pleiads Archived Website

Computers

MSX

• MSXAdvance

Sharp X1 Turbo

• X Millennium Advance

BBC Micro

• PocketBeeb

Miscellaneous

Sierra AGI

• GBAGI