Diminished

There are only three diminished chords, and each one has 4 inversions. you can take any note in a diminished chord and make that the root... which will give you another inversion

In Major, occurs on 7th scale degree
In Natural Minor, occurs on 2nd scale degree
In Melodic Minor, occurs on 2nd and 7th scale degree (which is a sharp 7)

A diminished triad can be thought of as a rootless dom7 chord

this is precisely the reason why diminished chords serving a dominant function aren't that common, as they are seen as a weaker version of a regular dominant chord.
ex. if we take a Bdim triad and treat that root as if it were a 3rd, then we add a G (which would be the new root), we end up with a simple dominant 7th chord

as a Migration Chord

Diminished chords are commonly used as "migration chords", whereby they will take us from one chord to another

Examples

Bennie and the Jets at "...that's been known to change the weather"
All Star - Smash Mouth at "...get your game on, go play"

Note: Even though our ears might interpret the migration chord as diminished, this is not always the case. It could be that what we're actually hearing is a major chord with the passing tone in the bass. Notice that there's one note difference between these 2 chords, and probably what you'll find is that the diminished version sounds too unstable

All Star seems to use the diminished chord, while in Bennie, it seems to be a major chord with 3 in the bass

as a Line Cliche

They are also used as line cliches, where we start with a more or less "normal" note, and move through a progression by dropping the bass tone.

to the ear, it seems all we are doing is just walking down the bass, but in reality we are creating a chord progression.

Examples

Life on Mars - David Bowie at the intro, the progression goes F - Am/E - A°/E♭

Diatonic

Although diminished chords appear at the 7th scale degree in the major scale, they are not often used like this.

the reason is because it's considered to be an incomplete version of the dominant chord.
- ex. in the key of C major, the 7th scale is a B°, and the dominant is a G⁷, which contains all the notes of the B°.

Variations

Diminished chords can exist in 3 forms:

Half Diminished vs. Fully Diminished

Half Diminished

1-b3-b5-b7

Often, half diminished chords are used instead of diminished triads. This is accomplished by adding a minor 7 to the diminished triad.

Fully Diminished (a.k.a Diminished Seventh)

1-b3-b5-bb7

note: A diminished 7th is enharmonically equivalent to a 6, so a fully diminished chord can be thought of as 1-b3-b5-6

A diminished triad with a diminished seventh added (ie. stacking minor thirds)

A fully diminished chord is symmetrical similar to diminished chords.

put another way, an augmented chord is what you get when you split an octave evenly in 3 parts, while a diminished seventh is what you get when you split it in 4 parts.
as a result of this symmetry, out of context from the key, each note of the chord could serve as the root.
Because diminished 7th chords are fully symmetrical, an Ab dim7 is the same chord as B dim7, D dim7, and F dim7 — just different inversions. So there are really only three distinct diminished 7th chords in existence

Note: the relationship between a fully diminished chord and the seventh chord 2 whole steps down from it (e.g. a fully diminished D# and a B7). The only difference between these 2 chords is that the diminished 7th is flattened, which now serves as the root for the seventh chord.

Diminished sevenths are symmetrical, made of all minor thirds, so anything can be the root.

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