I do a lot of Fox vs. Sheik. Here's my theory.
Final Destination - Even.
Fox:
1. Can up-throw --> up-air without platform interruption.
2. Has plenty of space to laser camp.
3. Edgeguards Sheik pretty well due to lack of platforms.
Sheik:
1. Dash attack leaves Fox WIDE open for a free hit. No escape at most %'s.
2. Has an easier time creating space for a full-needle hit (almost 20% damage)
3. Has a really harsh tech-chase game against Fox after d-throw.
Shiek can hold her own against Fox even with his speed if she positions herself correctly. On any stage, Sheik can **** Fox at low %'s if she gets him off the ledge so the large stage is a plus for Fox. I've played this match-up so many times and man, it can go anyway depending on the person's performance.
Pokemon Stadium - Fox's Advantage
Fox:
1. Has many good positions to be in when the stage changes. Stage changes usually require a vertical kill which Fox has the advantage in.
2. When Fox is recovering, if he does not aim for the ledge he has platforms, windmills, and even another ledge on a mountain to escape to when landing. Sheik does as well, however the lag from her up+b makes it easier to punish.
3. MANY places where he can drillkick infinite including four+ places around the mountain, three areas around the tree, and that weird **** on the right side when the mountain stage is in place.
Sheik:
1. Can escape up-throw --> up-air combos better depending on the level of the stage.
2. Can find places to "hide" in order to charge needles.
Fox's best stage in my opinion. Sheik can have an awkard time trying to recover from the stage. This isn't to say that Sheik can't beat Fox here, but yeah. W/e. Better stage for Fox.
Battlefield - Sheik's Advantage
I won't do a huge synposis. Sheik edgeguards REALLY well here since Fox can't stage climb his up+b. The small stage makes it easier for Sheik to toss Fox off the ledge. Ultimately, Sheik's best advantage over Fox is her edgeguard. This is the perfect place for her to use it. Any advantage Fox may have over Sheik here simply doesn't match Sheik's.
Yoshi's Island - Fox's Advantage
Yeah, **** that format I had earlier. I'm at work and takes too long. The super low ceiling is just way too good for Fox. Basically, Sheik's ultimate advantage on this stage is the small playing level which = edgeguarding, but Fox can get interference with the help of Shy Guys, a cloud, and the platforms. The super low ceiling makes it a lot easier to kill Sheik, which in my opinion, overshadows Sheik's advantage.
Dream Land - Even.
Whoever plays better wins here. Neither of the two have any real major advantage on this stage.
Fountain of Dreams - Sheik's Advantage.
The moving platforms benefit Sheik the most. A dash attack onto one of these suckers has a good chance of being followed with a slap which has a good chance of pushing Fox off the stage which has a good chance of Fox dying. The fact that the stage is largest underneath it makes it perfect for Sheik to get gimpy kills. Fox can up-smash from underneath ledges, and the low ceiling is a plus buuuuuuut I still think Shiek's got it better here.