I can remember back in 2002, when I first joined the Tape Op team as Reviews Editor, ribbon mics were relatively rare, and even new ones, like the now ubiquitous Royer Labs R-12, weren't too often seen in project studios like mine. Thanks in part to David Royer and Wes Dooley (AEA), two untiring champions of high-quality ribbon mics, and thanks also to several companies that created a market for low-cost mics of this type, ribbons are much more popular these days.
While slowly building up my collection of ribbon mics, I've used one or more ribbons on nearly every tracking session. Most ribbon mics have a figure-8 polar pattern, so they pick up as much sound from the back as they do the front, and they pick up nearly zero from the null points on their sides. Also, ribbons tend to exhibit strong proximity effect at close distances, but beyond a certain threshold (which is particular to the model of mic) the voicing stays fairly constant as mic'ing distance is increased. There are some obvious uses that take advantage of these principles. For example, take two singers and put them on opposite sides of a figure-8 ribbon and have them sing together. Or, use two ribbons next to each other in Blumlein configuration to mic a singer-songwriter — one ribbon on the voice, positioned so that the acoustic guitar is in the mic's null, and the second ribbon on the guitar, with the voice in the null — to maintain separation and phase-coherence.
Ribbon mics exhibit several more traits that you can use to your advantage. The first is terrific transient response, because the ribbon material is very thin and very light. The second is forgiving high-frequency reproduction, because the low-mass ribbon is less susceptible to overshoot and ringing at frequencies that can over-accentuate edgy, brittle, or sibilant tones.
That's why I love to use ribbons on drums. I want punch and detail throughout, while minimizing harshness from the cymbals and high-hats. For example, try placing a ribbon several feet from the drum kit. Think of this mic's position in radial coordinates with the kick drum at the center, and move the mic in/out to change the blend of drums and room ambiance, and radially to vary the relative balance of the individual drums. Compress to taste. (Don't be afraid to move the drum kit if you're not happy with the room ambiance.) Yes, you can try this with a figure-8 condenser mic, but I find that most condensers tend to lose way too much bottom end when you move them outward, and adding a "side" or "front-of-kit" condenser mic like this to the other drum mics can make cymbals sound too strident.
I also love using ribbons on vocals — particularly large ribbons on singers who are sibilant or overemphasize consonants. At distances greater than what you would typically use with a condenser or dynamic, a good vocalist can "work" the ribbon mic to control the blend of direct voice and room ambiance, without the voice becoming too thin or losing detail.
In the same manner, ribbon mics are also great for percussion. Most percussion instruments sound terrible to me when they're direct-mic'd "dry." Instead of setting up two or more mics (direct and ambient), I like to use a single ribbon. Again, I'll position the source radially to the mic to vary the roominess and "depth" of the recording. Handclaps especially sound most "real" to me when multiple people are clapping several feet from the mic, within the two arcs close to the mic's nulls.
A common theme in all these suggestions is that moving a ribbon mic can impact the sound significantly. Try it. It's much more fun than reaching for an EQ or a plug-in!