The Rules
Here, I'll explain the rules that pertain to all of the games in general; note that some of these rules are not set in stone, and may be subject to change.
You may use only your designated type and dualtypes that share the chosen type in battle.
If the chosen type is a type other than the one of your starter, you must use only your starter until you find a desired pokemon of your designated type; you must then ditch your starter.
If your starter or some other pokemon does not have the chosen type but gains it through evolution, you may use that pokemon. (i.e. Nincada into Ninjask or Charmander into Charizard for a Flying-mono)
HM slaves are allowed, but you may not use them in battle under any circumstances. Pickup is to be used at your own judgement; personally, I would use only one Pickup user.
You may trade for evolution, but you may not trade for pokemon not found in your game or extra TMs of any kind. You may use only the TMs found on a single cartridge.
If and when you reach an area where there are pokemon exclusive to your game's counterpart, such as Weedle in Viridian Forest in Yellow, Electabuzz in it's area in LeafGreen, but aren't found in your game, you may trade for these pokemon.
Cheat codes of any kind are not allowed, but glitches are, to a reasonable extent.
I believe that's about it. Now, onto the games themselves....
Red/Blue/Yellow
Ghost, Ice, and Dragon monotypes are not allowed due to the lack of those types.
The challenge is considered complete upon the reaching and defeat/capture of Mewtwo.
Psychic, Bug, Grass, Rock, Fighting and Electric monotypes are also discouraged for use due to a lack of variety; Mew is discouraged since it is technically not supposed to exist without an event that is long gone.
In the case of a Flying mono challenge, you are allowed EITHER Dragonite or one of the legendary birds. In their respective Fire, Electric and Ice mono-challenges, the legendary birds are allowed.
Gold/Silver/Crystal
The challenge is considered complete upon the defeat of Red/Ash.
Ghost, Dragon, Dark and Steel monotypes are not allowed due to a lack of pokemon.
In the case of a Flying mono challenge, you are allowed EITHER Dragonite or one of the legendary birds. In their respective Fire and Psychic monotypes, Ho-Oh and Lugia are allowed.
Celebi, however, is discouraged. I'm aware that it can be obtained in Crystal through the Celebi trick, but it's not supposed to be obtainable in the game apart from a very old event.
The wandering legends (Suicune, e.t.c) are allowed in their respective monotypes.
Ruby/Sapphire/Emerald
In Ruby/Sapphire, the game is considered complete upon the defeat of the Elite Four and the champion, and the reaching and defeat/capture of Rayquaza.
In Emerald, the game is considered complete upon the defeat of Steven. Capturing or defeating Groudon and Kyogre is also an optional goal.
Fire, Ghost and Dragon monotype challenges are not allowed due to a lack of pokemon.
In the case of a Flying monotype, you are allowed EITHER Salamence or Rayquaza.
Steel monotypes are not allowed unless you have Jirachi from the Colosseum bonus disc or something.
FireRed/LeafGreen
The challenge is considered complete upon the reaching and defeat/capture of Mewtwo and the defeat of the second wave of the Elite Four.
In the case of a flying monotype, you are allowed EITHER Dragonite or one of the legendary birds. In their respective elemental monotypes, the birds are allowed.
Pokemon caught in the Sevii Islands are allowed, and indeed necessary for some monotypes.
Steel, Ghost, Dragon and Dark monotypes are not allowed due to a lack of pokemon.
Diamond/Pearl/Platinum
The challenge is considered complete upon the reaching/defeat of Heatran, although obviously you may continue fight and level up your rival.
Dragon and Fire monotypes are not allowed due to lack of pokemon.
Additionally, Ghost, Steel, Rock, Electric and Dark monotypes are discouraged due to the lack of variety before the National Dex-by that time, you're near the end of the challenge, anyway.
In Platinum, due to the expanded Sinnoh dex, Steel, Electric, Dark and Ghost monotypes are allowed.
lv. 31, butts, lax/compoundeyes: psybeam, gust, leach life, sleep powder
lv. 30, tushy, sassy/early bird: headbutt, mach punch, focus blast, silver wind
lv. 27, condom, lax/hyper cutter: brick break, revenge, dig, strength
lv. 27, boobies, rash/swarm: brick break, shadow claw, aerial ace, rock tomb
lv. 28, phallus, relaxed/swarm: twineedle, poison sting, rock smash, pursuit
as my starter (since it evolves into Fire/Fighting). Let's see how it goes.