The main problem you run into with short scale guitars is they're always going to have quite soft note attack and decay, and then people go and whack gargantuan strings on them and lose all harmonics, so all you're left with is a spongy fundamental that sounds like you plugged into a bass amp with a towel over the speakers. So my first bit of advice is don't try to compensate for the scale length any more than you would with a 24" guitar, or you may even want to treat it like a 24.75". Unless you're tuning very low, the idea that short scale guitars need heavy strings is a baseless myth brought about by people who spend too long theorising on forums and not enough time actually trying stuff out. Don't forget, Brian May used to use .008-.034 strings on his 24" guitar for a couple of decades and only recently moved up to .009s, and nobody other than the most blinkered bedroom deathmetalhead is going to say he sounds bad. Billy Gibbons, pretty much the same deal, has used .008s and .009s on 24.75" guitars for years and his tone can be extremely full when he wants it.
And to expand that to a second point, don't forget that it's easier and cheaper to change string gauge or material later than it is to change pickups. You're always better off getting a pickup that meets your basic requirements—humcancelling or not, power, rough tonal EQ—and trying it with strings that are just one step (or even just a half) heavier than what you'd use on your regular guitars. Then if that works, great, or if it does sound a bit too wimpy then you can just spend £8 on another pack of strings to fix it, with a better idea of what kind of gauge you need to jump to now you've at least tried it with a standard or near-standard gauge. Better that way around than setting on a particular string gauge and then trying to compensate with the pickups, potentially spending ten times more and still getting it wrong.
All that said, I've found in 22-24" guitars my biggest enemy is sustain, so I now favour weaker magnets and/or pickups that have the magnets at the bottom (e.g. humbucker or P-90 style) rather than right up to the strings (e.g. firebird or strat style). AlNiCo 3 magnets have proven to be pretty great for this, with a couple of my short guitars having a noticeable improvement in sustain just from swapping the magnets in their pickups from A5 or ceramic to A3. Of course A3 also produces less output and a fractionally warmer and more open tone, but if that's not wanted it's easy to compensate for that by changing control pot values (e.g. 1meg instead of 500k to bring back some output, especially in the high-end) or by just adjusting your amp's EQ a little. (Budging the gain and mid controls up by 1 does wonders for an A3 pickup.) If you go for any kind of humbucker tone (series or parallel) make sure you're working with either Strat/Tele-sized humbuckers or asymmetrical coils in a full-size humbucker, since any full-size humbucker with matched coils is going to struggle with clarity in short scales.
To that end, for humbucker tones I go for a SD Hot Rails, Cool Rails or Dimarzio Pro Track, while for single coil-ish tones I go for a SD Vintage Rails or DM Chopper, all depending on the rough output I want. Then I shove in A3 magnets instead of the ceramics and call it a day; any further tone shaping happens at my amp. I avoid using full-size humbuckers both since space is often at premium and because they get a lot muddier on short scale guitars, especially at the neck. For 'true' single coil tones I go for the DM Area 61 or Virtual Vintage 54 Pro—whatever happens to be in stock/cheapest at the time, I've never actually been able to hear a difference between them—or if I'm feeling fancy I'll try to use older Fender Vintage Noiseless telecaster pickups, the ones made with bar magnets at the bottom rather than magnetic pole pieces. A couple of times when needing to slap something in quickly I've bought Squier Classic Vibe A3 strat pickups (I believe they're rebranded Tonerider Surfaris) and those do okay as a basic single coil tone in a pinch with minimal string pull, though I've always ended up eventually replacing them with the hum-cancelling designs mentioned before.
The Hot Rails, Cool Rails and Pro Track all do well with series-parallel switches, so if you want flexibility I highly recommend those. Don't bother splitting any of the aforementioned pickups; the coils are too thin and it sounds like a cheap radio. My 'default' configurations these days are a Hot Rails and a Vintage Rails for two-pickup guitars with the Hot wired to a series-parallel switch, or Cool/Vintage/Cool with the two cools both wired to a single series-parallel switch (requires a 4pdt, bit hard to find in some countries but they are out there).
However, something that can throw all of that out the window is active designs. I've found active pickups work great in short scale guitars, as unpopular as that thought may be to the ska and grunge short scale purists. Most active pickups (at least EMG and Seymour Duncan designs) have low string pull and many have wide response ranges so you don't have to worry about scale length or string gauge with them, they can cope with whatever. The only thing that prevents me from sticking EMG 60AXs, 89Xs and SAXs in all my short scales is needing space for the battery and, for some active pickups, slightly deeper pickup routings which short scales almost always lack. Still, have a look inside the guitar and see if actives will fit, because as long as there's enough space they're the most straight-forward solution to short scale balancing. They're especially good when you want good response but no specific set-in-stone tone; check out Prince at the Montreux jazz festival in 2009 for an example of the huge variety of tones that can be worked out of a simple 85+SA set.
