I don't use multiband compressors anymore, having replaced them with more-versatile dynamic equalizers.
If you set the threshold low and apply a gentle knee, multiband compressors are good for leveling, especially on vocals and bass.
It can help mitigate acoustical issues, such as a vocal tracked too close to a wall. Similarly, it helps with inconsistent volume on a bass track, especially if the bass cab is miked.
You can use it as a de-esser when the offending sibilance isn't neatly contained within a single band. Along the same lines, it can be used to compensate for proximity effect when a singer has occasionally gotten too close to the mike.
I don't use it this way, but many do: glue compression on the master bus, where a multi-band can avoid pumping.