As much as Riot tried to say "skins won't bring any advantages", the fact is there are skins that have an edge over their original one (and other skins). Most of what I list here bring rather small advantages, yet, they are easy to notice and any advantage, even the smallest can help. Some guy on youtube already made a video about this, but his video is largely inaccurate, and many skins that he listed even bring small disadvantages for the users.
For example, when your champion model looks just a bit smaller, the hitbox is the same, so the enemy will aim more carefully, and have a higher chance hitting you (because even if they miss your model by a bit, they in fact will still hit you due to the same hitbox). However, when your champion model looks so much smaller, it is actually harder for the enemy as they really struggle to aim, and as they often need to predict your movement while firing, being much smaller than normal actually becomes an advantage. Currently I find only 1 skin in the league to "qualify" for this.
On the other hand, if your skill looks smaller, the enemy will just move a bit out of the animation, thinking he has dodged it, but in fact the hit box of the skill is the same (which is larger than the animation) so it will hit them.
I try to list some skins that bring advantage to their user. I exclude any skins with sunglasses that receive 1 less damage from Leona's passive. That thing is, to me, too minor to be mentioned. Feel free to add anything you find impactful that I've missed.
- Blackfrost Anivia: much more reliable AA. Even with recent buff to her AA animation, this skin still proves much better at last hitting.
- Frostfire Annie: harder to see the stun when the shield is on.
- iBlitzcrank: the grab looks really thin and the tether is almost invisible the further the grab travels, making it harder to dodge.
- Dragonwing Corki: better AA (I like Urf, but his Corki skin AA is simply too bad and distracting).
- Santa Gragas: Tiny barrel with the same hit box. Careless opponents that move just a bit far away from the barrel will still get hit.
- Forecast Janna: better AA (though rather less impactful as she is mostly go support).
- Mecha Kha'Zix: in addition to the visual, there's sound to indicate when the champion is isolated.
- Prestigious LeBlanc: she is SO MUCH smaller than her original.
- Steel Legion Lux: famous as "the buggiest skin of all time". Both Q & R have smaller animations, and Q is visible only from the end of her staff resulting in 1/5 second less time for any opponent to dodge it.
- Darkflame Shyvana: when using R from fog of war, with W on, she is very hard to see for the 1st 1/4 second.
- Rocket Girl Tristana: better AA animation than the original =>>> better last hitting.
- Underworld TF: somewhat faded Q. (more ARAM advantage than SR).
- Archlight Varus: in the same vein as Steel Lux skin. A buggy Q with much larger hitbox than the shown animation.
- Pool Party Ziggs: the Q bounces look much more irregular.
- Snowday Ziggs: only true for ARAM. His skills are MUCH harder to see (and dodge).