Ribbon microphones sound full, rich, and smooth. They're more detailed than moving coil dynamic mics, yet less harsh in the treble range than condensers. Do your recordings sound cold, scooped or somehow "digital"? Try a ribbon microphone! Many leading engineers such as Bruce Swedien are certified ribbon enthusiasts.
Ribbons have always been popular for vocal and spoken word recordings. Ribbon mics produce that certain "larger than life" sound. They make voices sound smooth, big and elegant. A ribbon mic is an excellent choice for slightly harsh voices. Moreover, most ribbons are pleasantly unproblematic regarding sibilants (S-sounds).
Another typical ribbon domain is strings and brass; woodwinds, too. Ribbons lend a certain gloss and lushness to these instruments. Think of Hollywood soundtracks from the 40s and 50s - those were recorded with ribbon microphones almost exclusively. Many engineers still prefer ribbons over condenser mics when it comes to capturing a full, slightly "romanticised" orchestral sound.
Ribbons are also great for drums and percussion. Their harmonic sound image lends a certain colour to these instruments and unites them to a homogeneous instrument group. If you have a problem with harsh, loud and too-prominent cymbals, try ribbons as overheads!
More recently ribbons have been discovered as the ultimate guitar amp microphones. Ribbons typically have a very even midrange. Electric guitars are actually nothing but midrange. The frequency range of a guitar speaker is usually 100-7000 Hz. That's it! But that limited frequency range happens to be the preferred area for the human ear, so it's easy for us to discern even miniscule variations or colorations in that range. Ribbons are able to capture even the finest nuances of guitar sounds. Your guitars and amps will sound as expensive as they actually are. Well, maybe even more expensive!
What's true for all microphones is even more true for ribbons: experiment! Let your ear be your guide; there are many more applications to be discovered for ribbon microphones!