About Fm/C — uncommon voicings
Fm/C is C, F and Ab — with C put in the bass.
It sounds like the minor triad it is, resting on a different floor. If C is already one of the chord’s notes this is an inversion, and if it is not it is a new colour under an unchanged chord; either way the reason to reach for it is a bass line that walks by step where root-position chords would leap.
Inversion or pedal — the harmony is unchanged; only the bass moves.
Also written Fm over C or Fm with C in the bass.
What these less common shapes are
How to use Fm/C
In which keys. Every note of Fm/C belongs to C minor (the iv), C# major (the iii), Eb major (the ii), F minor (the i), Ab major (the vi) and Bb minor (the v).
What to play over Fm/C
Every note of Fm/C is in F minor pentatonic, F blues, F natural minor, F dorian and F phrygian.
Each of those is a page: the scale across the neck, and the chords it harmonises into — which is where Fm/C sits in it. Every scale.
Chords near Fm/C
Change one note of Fm/C and you have (the minor third becomes the major third), (the fifth becomes the flat fifth), (the minor third becomes the second), (the minor third becomes the fourth), (add the sixth) or (add the flat seventh).
Fm/C is a slash chord — a triad over a named bass note. Slash chords on every root.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.