*Note that I will not list most of the legacy skins; However, I will list legacy skins that were given out during promotions.
All Star Akali - 975 RP (Limited - Football World Cup in RPA 2010)
Black Alistar - (Limited - digital pre-order)
Vancouver Amumu - (Limited - Winter Olympics in Vancouver 2010)
Regifted Amumu - 520 RP (Limited - Snowdown 2010)
Team Spirit Anivia - (Limited - Winter Olympics in Vancouver 2010)
Goth Annie - Included in Digital Collector's Pack (3900 RP)
FrankenTibbers Annie - 975 RP (Limited - Harrowing 2011)
Rusty Blitzcrank - (Limited - discontinued 17.01.2010)
Goalkeeper Blitzcrank - 520 RP (Limited - Football World Cup in RPA 2010)
Definitely Not Blitzcrank - 520 RP (Limited - Harrowing 2011)
Arctic Warfare Caitlyn - 975 RP or PC Gamer Promo, North America (If you didn't get this from the promotion, you can still buy it)
Nightmare Cho'Gath - 520 RP (Legacy ; Used to be free through partnership with Alienware)
UFO Corki - Free for accounts created before 14.01.2010 (Limited)
Ice Toboggan Corki - (Limited - Winter Olympics in Vancouver 2010)
Mundo Mundo - 975 RP (Limited - Harrowing 2010)
Striker Ezreal - 520 RP (Limited - Football World Cup in RPA 2010)
Union Jack Fiddlesticks - (Limited - Winter Olympics in Vancouver 2010)
Pumpkinhead Fiddlesticks - 520 RP (Limited - Harrowing 2010)
Toy Soldier Gangplank - 975 RP (Limited - Snowdown 2011)
Santa Gragas - 975 RP (Limited - Snowdown 2010)
Snowmedinger - 975 RP (Limited - Snowdown 2011)
Victorious Jarvan IV - Limited: Gold or Platinum medal in Season 1
The Mighty Jax - (Limited - Winter Olympics in Vancouver 2010)
PAX Jax** - Limited: Pax 2010 or Pax East 2011
Festival Kassadin - (Limited - Winter Olympics in Vancouver 2010)
Red Card Katarina - 975 RP (Limited - Football World Cup in RPA 2010)
Kitty Cat Katarina - 975 RP (Limited - Harrowing 2010)
Silver Kayle - Collector's Edition
Judgement Kayle - Limited: 10 matched games in Season 1
Reindeer Kog'maw - 975 RP (Limited - Snowdown 2010)
Festive Maokai - 975 RP (Limited - Snowdown 2011)
Candy Cane Miss Fortune - 975 RP (Limited - Snowdown 2010)
K-9 Riot Nasus - Meet Rioter personally
Snow Bunny Nidalee - (Limited - Snowdown 2009)
Bewitching Nidalee - 975 RP (Limited - Harrowing 2011)
Haunting Nocturne - 520 RP (Limited - Harrowing 2011)
Workshop Nunu - (Limited - Snowdown 2009)
Lollipoppy Poppy - 975 RP (Limited - Harrowing 2010)
Ragdoll Poppy - 975 RP (Limited - Snowdown 2010)
King Rammus - Limited: Closed Beta Testers
Human Ryze - (Limited - Amazon/Gamestop Pre-order)
Champion Ryze - Win certain LoL Events
Zombie Ryze - 975 RP (Limited - Harrowing 2010)
Nutcracko - 975 RP (Limited - Snowdown 2010)
Riot Squad Singed - Limited: Meet Rioter personally (before GamesCom 2011)
Huntress Sivir - Direct 2 Drive or Gamer's Choice Pack
PAX Sivir* - Limited: PAX Prime 2011 Goodie Bags
Silent Night Sona - 520 RP (Limited - Snowdown 2010)
Happy Elf Teemo - (Limited - Snowdown 2009)
Earnest Elf Tristana - (Limited - Snowdown 2009)
Viking Tryndamere - 975 RP or promotion code from Dreamhack Summer 2011
PAX Twisted Fate - Limited: PAX 2009
Whistler Village Twitch - (Limited - Winter Olympics in Vancouver 2010))
Medieval Twitch - 50 referred friends with RaF 2.0 (350 with RaF 1.0)
Curling Veigar - (Limited - Winter Olympics in Vancouver 2010)
Nosferatu Vladimir - 975 RP (Limited - Harrowing 2010)
Grey Warwick - 25 reffered friends with RaF 2.0 (50 with RaF 1.0)
Urf the Manatee - 5000 RP (99% sales during first week) (Limited - April's Fool 2010)
Winger Hussar Xin Zhao - 975 RP or help with bug testing Polish client version
Old St. Zilean - (Limited - (23.12.2009 - 27.12.2009))
I have been to PAX East 2012 this year, and have gathered some knowledge about the PAX skins.
*PAX Sivir could be obtained @Riot Games Booth at PAX East 2012.
**PAX Jax was difficult to obtain at PAX East 2012, but was still available then. I had to win it from this "Spin to Win" wheel of fortune type game there.
Best Answer
As mentioned, the only thing skins change is the appearence. No stats are affected nor supposed to be. If they were, people would fully classify lol to be a pay2win game, which few does at the moment.
One thing to note though, is that some skins can change the appearence of their spells. I've noticed some times that the abilities are actually harder to spot by the eye on certain skins.
An example is Pulsefire Ezreal, whom in my opinion is harder to track some of his abilities on. His skill shots gets a bit harder to see exactly where the hit area is.
I can aswell imagine Frostfire Annie and her abilities to be a bit more hidden during the winter version of Summoners Rift.
A clear difference is the Nocturne original and Nocturne halloween skin. The Nocturne halloween skin's Shadow ability almost cuts through your eyes due to its strong colors. It might appear to be a bad thing, but can be useful to drag attention with when you're going tanky with him.
But basicly, the only thing skins might improve your gameplay with is visual advantage to a low extent, and the feeling that you must win because, hey, you got a skin !
EDIT
It seems as if Singed's poison had been purple all along. Never mind this case: