Black hole and main sequence star/giant star
We can observe binary systems containing a black hole by looking for emissions from accretion disks which may form when matter is transferred from the companion star. X-ray binaries and microquasars are particularly notable types.
The compact object doesn't have to be a black hole - neutron stars also sometimes show up - but in some cases, measurements indicate that the remnant is a black hole. In other cases, either the mass of the compact object is poorly constrained or could still be a massive neutron star, so for many systems, the nature of this object remains unknown.
- Cygnus X-1: The companion star here is HDE 226868, a massive O-type star.
- V404 Cygni: The companion star is a K-type star (much less massive than HDE 226868); the system forms a microquasar, with occasional high-energy x-ray emission.
Black hole and white dwarf
It seems a logical assumption that black hole/white dwarf systems should be possible, and indeed they are. Mass loss through accretion by a companion object can change a star's evolutionary future (see the Algol paradox for a particularly weird case), but for many systems with a black hole, the "normal" star should still progress through the main sequence and post-main-sequence evolution, possibly becoming a white dwarf.
Matter transfer can still happen at this stage, of course, so emission may continue, although the nature of the radiation could be changed. It's true that there aren't many known systems of this type, but there almost certainly more.
- X9: The companion star is a white dwarf orbiting extremely close to the black hole. Bahramian et al. are confident that the companion is a white dwarf, and the main object is likely a black hole.
Black hole and neutron star
These type of systems are possibly the most interesting from a gravitational point of view. They should produce gravitational waves, which could be detectable by LIGO.1 LIGO has yet to observe such a system, which seems a bit puzzling, although something like the blue supergiant in Cygnus X-3 could evolve into a neutron star in the future. This is a case where more data and observations are needed.
1 You've probably heard the rumors that LIGO detected a possible neutron star-neutron star merger, which would imply that its sensitivity to neutron star-binary systems could indeed be high enough for detections, as expected. It turned out that it was simply a binary black hole system - not quite as exotic as some people were hoping for.