My main problems have pretty much all been covered here, so this is probably just recap.
There's the problem that too few skills have scaling rewards past the minimum success.
Solution: Skill checks need to have extra benefits for exceeding the DC, whenever it makes sense to add them.
Plus, DCs in the current system are either trivial if you're trying, or impossible if you're not (note that "not trying" might even include having max ranks). Competence items that scale up to +30 are a major culprit here, but they're not the only bullshit bonus.
Solution: Reign in miscellaneous skill bonuses to compact the spread, and assign DCs based on what actually gets accomplished, not what level you think people should be doing things at. At the same time, take into account the actual spread of the RNG relative to real people; DC 20 isn't "nearly impossible", because even 1st level commoners will have bonuses if they have any training. At the same time, DC 40 isn't the DC you should assign to climbing a mountain because you don't want people crossing it until past level 10 or something, or because you think it's really hard so only really awesome (and thus high level) characters can do it.
Then there's the problem where not all the skill divisions make sense; Use Rope on its own is kind of weird considering how broad Survival is. Plus there's the one with Skill taxes (you need both Hide and Move Silently) for certain niches.
Solution: Fold various skills into one another. This also helps people diversify in skills, since they now have more points relative to the number of skills that exist.
There's no reason to use most skills proactively, either; 9 times out of 10, you make a skill check in response to something the DM throws at you. The exceptions are mostly social interaction or specific schticks that other resources (like feats) enable, and a few one-off checks that you might make once per campaign if you're lucky. So there's not much of a huge incentive to pump your ranks in Jump, except possibly as part of some other schtick; it has very little value in itself.
Solution: Have skills grant actual benefits in a wide variety of situations, not just "You need to pass this to get to the plot" situations. Athletics increasing movement speed and suchlike.
Class skills always bugged me since I've come around to dissociating class with a character's fluff. There's no reason the dude who's memorized where kidneys are for stabbing can't have full ranks in Heal or Knowledge (nature) or what have you.
Solution: Disentangle skills from classes. A class can still grant skill points, if you want, but everyone should have some number of class skills, freely chosen, so as to suit their character background better.
Skills themselves rarely have a lot to offer, except for a few that everybody takes as a result when they can. Skills like Use Magic Device, Iaijutsu Focus, or any of the Knowledges that identify creatures, are the exceptions that make it impossible to fit other skills onto a character if they're not absolutely necessary, which is just bad for helping validate as many archetypes as possible.
Solution: Make more skills actually useful. This goes along with the problem of skills not having a proactive use; you need to give people a reason to take ranks in skills. This is a good place, by the way, to have ranks-based benefits, to incentivize actually training instead of just grabbing a mystical doodad or whatever. Ideally, you should have ranks-based benefits, and DC-based benefits; the former probably passive, like a bonus to speed, the latter active, like the ability to increase your initiative count slightly or something.