Temporary bit here, anybody who wants to contribute links to guides for the modded solutions, please do so, I'm going to be searching myself.

Alright, this is a guide in progress, I am by no means an expert, so if you feel I am wrong about any of this information, please tell me.

I am an American, so I only have experience with NTSC, if any of you want to contribute PAL information, I'd be happy to add it. I am going to try to cover the best non-modified solutions first, and add in mods later, as I like to keep my hardware virgin.

Atari 2600

Best non-modded solution: RF
Best modded solution: S-Video

Atari 5200

Best non-modded solution: RF
Best modded solution: RGB VBXE Board
Second best modded solution: S-Video

Atari 7800

NTSC best non-modded solution: RF
PAL best non-modded solution: RGB (French 7800s only)
Best modded solution: S-Video

Bally Astrocade/Professional Arcade

Best non-modded solution: RF
Best modded solution: S-Video

ColecoVision

Best non-modded solution: RF
Best modded solution: Component

Emerson Arcadia 2001

Best non-modded solution: RF
Best modded solution: Composite

Magnavox Odyssey 2

Best non-modded solution: RF
NTSC best modded solution: Composite
PAL best modded solution: RGB (only PAL consoles)

Mattel Intellivision/Intellivision II

Best non-modded solution: RF
Best modded solution: Composite

Sega SG-1000/SG-1000 II

Best non-modded solution: RF
Best modded solution: RGB (later SG-1000 IIs only)
Second best modded solution: Composite

Famicom

Best non-modded solution: RF
Best modded solution: RGB (requires replacement PPU)
Second best modded solution: Composite

Nintendo Entertainment System

Best non-modded solution: Composite
Best modded solution: RGB (requires replacement PPU)

Sega Mark III

Best non-modded solution: Composite
Best modded solution: RGB Present in A/V, need amplification
Second best modded solution: Component (video encoder outputs component video but no mod has been found yet)

Sega Master System

PAL Best non-modded solution: RGB
NTSC best modded solution: S-video

Sega Master System II

Best non-modded solution: RF
PAL Best modded solution: RGB
NTSC best modded solution: S-Video

Nintendo Entertainment System 2

Best non-modded solution: RF
Best modded solution: RGB (requires replacement PPU)
Second best modded solution: Composite

Super Nintendo Entertainment System/Super Famicom

Best non-modded solution: RGB
Second Best non-modded solution: S-Video

Sega Genesis/Mega Drive

Best non-modded solution: RGB Model 1/RGB Model 2
NTSC best modded solution: S-Video

Super Nintendo Entertainment System Mini/Super Famicom Jr.

Best non-modded solution: Composite
Best modded solution: RGB
Second best modded solution: S-Video

Victor/Sega Wondermega

Best non-modded solution: S-Video
Best modded solution: RGB

Neo Geo AES

Best non-modded solution: RGB
Best modded solution: Component
Second best modded solution: S-Video

Neo Geo CD

Best non-modded solution: RGB
Second best non-modded solution: S-Video

Atari Jaguar

Best non-modded solution: RGB

TurboGrafx-16/PC Engine

Best non-modded solution: Composite (with TurboBooster, TG-CD or custom cable)
Best modded solution: RGB/RGB
Second best modded solution: S-Video

TurboDuo/PC Engine Duo/Duo-R/Duo-RX

Best non-modded solution: Composite
Best modded solution: RGB
Second best modded solution: S-Video

SuperGrafx

Best non-modded solution: Composite
Best modded solution: RGB
Second best modded solution: S-Video

3DO

Best non-modded solution: S-Video
Best modded solution: RGB

Amiga CD32

Best non-modded solution: S-Video
Best modded solution: RGB

Sega Saturn

Best non-modded solution: RGB
Second best non-modded solution: S-Video

Nintendo 64

Best non-modded solution: S-Video
Best modded solution: RGB/RGB (certain N64s only)

PlayStation

Best non-modded solution: RGB
Second best non-modded solution: S-Video

Dreamcast

Best non-modded solution: VGA for games that support 480p
Second best non-modded solution: RGB
Third best non-modded solution: S-Video

PlayStation 2

Best non-modded solution: VGA via SoG/Blaze/Xploder bootdisk for games that can be forced into 480p
Second best non-modded solution: Component
Third best non-modded solution: RGB
Best modded solution: Universal VGA cable

GameCube

Best non-modded solution: Component for games that support 480p
Second best non-modded solution: S-video
PAL best non-modded solution: RGB

Xbox

Best non-modded solution: Component
Second best non-modded solution: RGB
Best modded solution: VGA Sync-On-Green Cable and modded BIOS

Xbox 360

Best non-modded solution: HDMI
Second best non-modded solution: VGA
Third best non-modded solution: Component

Nintendo Wii

Best non-modded solution: VGA
Second best non-modded solution: Component
PAL third best non-modded solution: RGB

PlayStation 3

Best non-modded solution: HDMI
Second best non-modded solution: Component
Third best non-modded solution: RGB

If you have any information or corrections you'd like me to add, just post below, or drop me a PM.

Edits for 2/15/11:

Added Atari 2600 and 5200 (Thanks CRTGamer)
Added PAL information for SNES and Dreamcast (Thanks Hateshinai)
Added basic Master System information, I am not sure of it, so will some experts please chime in? (Thanks Scooter, I'll add the other Sega systems later on)
Formatting changes
Clarification and correction

I only own the systems listed in my signature, so I don't have much experience with the rest, but I'll be researching the rest to the best of my ability.

Edits for 2/16/11:

Added Atari Jaguar, TurboGrafx-16, and Panasonic 3DO (Thanks Segatari2002)
Added Atari 2600 solution (Thanks Hobie-wan)
Added Genesis information (Thanks cyborc)
Added Saturn, PS1/2, Xbox, information, (Thanks Anapan)

As always, please tell me if I get something wrong, thanks to everyone who has contributed so far!

Edits for 2/17/11:

Removed RGB information for NTSC, it is possible to use RGB on our displays, but not really practical.

Edits for 2/27/11:

Added 7800 and Neo Geos (Thanks Segatari2002)
Decided to bold all the titles

Edits for 4/9/12:

Added corrections from ApolloBoy.

Edits for 7/11/12:

Added links to guides for some of the mods.

Edits for 7/12/12:

Added links for some more. (Thanks ApolloBoy)

Looks like this will be a great guide!

As in the Gamecube choices
Agree with your Dreamcast VGA choice, but maybe a mention of SVideo as a secondary connection for non VGA supported games or non VGA HDs?

Oh of course, I forgot about that.

Thanks. Another one or two:

Atari 2600
Non modded - RF
Modded - Composite

Atari 5200
Non Modded - RF (Auto Power Switch on 4 port)
Modded - SVideo

PAL SNES and Dreamcast both support RGB with no mods. The quality is better than S-Video.
Do American and Japanese Dreamcasts support RGB? I seem to remember using my Japanese DC with RGB but I'm not 100% sure since I usually use a PAL one (and VGA) instead.

What about Sega Master System, Genesis/Sega CD/32X and Saturn?

Also, keep in mind that not all games will display video with component cables on a PS2. For example, Ico will not output video when the PS2 console is connected with component cables. I would suspect the same in true for other older PS2 games.

Both my NTSC and PAL at 60Hz ICO works, 480i still works with the Component hookup.

PS2 fat or new slim? My fat won't output video with component connected with certain older games.

Both my fats work fine, I'm installing a PS2 game at the moment, so can't check which revisions I have. But I do have two versions, one with Firewire, one without.

Unless I hear someone else verify this, I'm just gonna go with CRTGamer on this, I have a PS2 with component cables, and I've never had an issue with any games. Might just be your TV, man.