I can't answer the poll because it varies for me. I always check what tech I could get from a GP, but I only use them for it if it will gain me an immediate strategic advantage that plays into my game plan. Otherwise...
Great Prophets - since the Warlords patch nerfed using them for Civil Service, I tend to use them for shrines, then settle later ones in the Wall Street city (which usually has a shrine). I find they don't usually pop for worthwhile techs anymore. Meditation? Thanks, I'll pick it up on my own...
Great Scientists - like agc28, I use the first one on Philosophy, especially if Taoism hasn't been founded, the next one for an academy in my science city, and the next one, if he appears in time, to help with Education towards winning the liberalism race. However, I've worked out that you get better results from settling GS in the science city rather than using them for additional academies, unless you have cities producing 60 research points per turn before multipliers are factored in.
Great Merchants - in the early or mid-game, I tend to use these guys (especially the free one from Economics if I get him) for trade missions to finance upgrades of my veteran units. Later ones get settled in the Wall Street city.
Great Engineers - I almost always use them to complete wonders. I think I've only settled one once, ditto for popping one for a tech.
Great Artists - I'm not fond of these guys unless I'm going for a cultural win. My first preference is to use them for a technology, but if it isn't anything I want, I'll settle them in a border city. I used to use them for great works (i.e. culture bombs), but I've been seeing information that culture per turn will bring better results than one big spurt of it; also, I find cities I've just captured are usually better off in revolt, since they're often surrounded by enemy culture until I capture the next city.
As for golden ages, I almost never use great people for those. The only exception is during a pursuit of a space race victory, when most of my cities are building space ship parts.