Also note that the DMG does call out for experience from noncombat encounters. The most basic level is traps - they have a CR, they grant experience for being "defeated", etc. They also call out social encounters as being worth experience, though not as much as combat because of its nonlethality.

As for raising mundane things if you've only done mundane things - that can be enforced by the GM. Just let your players know that "Hey, I want your advancement to make sense, so if you haven't done anything regarding baskets, please don't raise your basketweaving skill". Then in social encounters, give a bit of experience - maybe 5% of the exp needed for their next level for a basic encounter where they have a goal and achieve it - finding out the mayor's secret from the town drunk, who used to work for the mayor, for example - and scale up the exp as the importance of the encounter rises. Keeping track of different types of exp and what they can be used for is just plain annoying. Bookkeeping is never fun.