In my experience, the best way to find uses for obscure skills is to let the players come up with them, and just go along with their suggestions if they're halfway plausible. If your players are creative enough, as yours seem to be, they will come up with ways to use their oddball skills if you let them.
To make this work best, you'll need to let your players have some control over the setting. For example, if your player has picked Knowledge (Music) as a skill, deliberately introducing an enemy that's vulnerable to music would be cheesy. But if the player asks you if, say, the giant snake monster thing attacking might be entranced by playing music to it — sure, that's a totally valid and, moreover, cool[tvtropes] thing to try, and you should totally let them use a skill roll to find an appropriate tune.
(Of course, a trick like that shouldn't necessarily finish the encounter, even if it succeeds. Sure, the snake is charmed now, but what's it going to do when the music stops? And it probably won't be deeply enough entranced that you could just walk up to it and stab it, or that it wouldn't react to ranged attacks...)
Similarly, designing a puzzle whose solution is "weave a basket" would be silly. But if your players decide to set up an ambush in the woods, and one of them wants to weave a sniper platform out of some bamboo stalks (and why shouldn't there be bamboo stalks?), then yes, you should definitely let them use their Basketweaving skill for it.
Or maybe the players need to carry some water but have no bucket, and one of them suggests weaving a reed basket and lining it with clay? Great idea, and excellent problem-solving. And yes, there should be some reeds and clay available, unless you've got some very good reason why there wouldn't be.
As a real-life example, in one vaguely horror-themed Roll for Shoes one-off I ran, one of the players decided to pick Knowledge (Waterfowl) as their free starting skill (I gave every player one). They found several uses for it during the game (such as, IIRC, using a duck whistle to summon some ducks to distract an enemy), even though I didn't deliberately introduce any waterfowl into the scenario. Of course, Roll for Shoes rather encourages such shenanigans as part of the system, but that doesn't mean you can't apply the general technique in pretty much any system.
The point is, your players will come up with uses for their skills that you never would've thought of alone. And because the uses for those skills come from the players, rather than being shoehorned in by the GM, they'll seem cool and creative rather than awkward.
Ps. One potentially tricky part, especially if you haven't been letting your players do much of this before and they've gotten used to your GM style, is getting your players to start thinking of creative applications for their skills. Of course, some groups might have no problem with this, but others might need a bit of nudging to get out of their current rut.
One approach is just to bluntly tell your players, out of game, that you're going to be trying to make your game a bit more player-driven, and that if they come up with any cool out-of-the-box solutions or uses for their more obscure skills, you'll try to accommodate anything they'll suggest.
During the game, you can also try gently nudging your players whenever you think of some way they could apply their skills. For example, if the players in the last example above are going to just try carrying water in their hands, you could just mention offhand that, hey, there's some reeds on the riverbank and probably some clay too, and didn't one of you guys know how to weave a basket...? Don't make it seem like it's the only possible solution (or the "intended" one), but just throw it in as an idea that one of the characters could plausibly have, if it makes sense to the players.
This is not a technique you should overuse, but hopefully, after doing it a few times, your players will start to come up with such suggestions on their own. Then it's just up to you to make the suggestions work (or at least fail in interesting ways) whenever possible. You can also use the nudging technique if you notice that some of your players are applying their skills much less than others, to encourage the less proactive players to contribute more.