BPM stands for Beats Per Minute, so the note used in the tempo indication should have the duration of one beat.
In a simple time signature like 2/4, 3/4, or 4/4, you should use a crotchet note (quarter note), because this is the duration of one beat.
But in a compound time signature such as 6/8, or 9/8, or 12/8, a quaver (eight note) is the duration of a pulse (which has less emphasis than a beat), a dotted crotchet (dotted quarter note) has the duration of a beat, and so this is what should be used in the tempo indication.
In a 'larger' compound time signature such as 6/4 or 9/4 (again compound time signatures), the beat duration is a dotted minim (dotted half note).
Irregular time signatures are a bit more difficult because the duration of a beat is not constant.
[where S = strong beat, M = medium beat, w = weak (pulse)]
5/8 has the beat structure of
SwwMw (but could also be
and 8/8 is
These irregular time signature should probably be denoted with dotted crotchets because this is the duration of a strong beat. (The second option of 5/8 should use a crotchet).
But, it would also be reasonable to instead uses a quaver in the tempo indication and write a PPM (Pulses Per Minute) instead of BPM.