The essential scales for transverse ocarinas
A scale is simply a collection of notes that sound good together. Many different kinds of scale exist, and these are, in my opinion, some essential scales every ocarina player should know.
These are presented for an alto C ocarina, and can be transposed to attain the equivalent scales for ocarinas in other keys.
Major scales
C Major

F Major
Notice that in this scale the tonic is in the middle of the range with a good number of notes both above and below it.

G Major
G major on a single chambered ocarina is an interesting one to consider as only one G exists on the instrument.

D major

Minor scales
The name 'Minor scale' could imply that they are somehow less important, but no, minor scales are just another kind of scale, formed from a different pattern of intervals.
D minor
The minor scales are based on the same pattern as the major scales, as if you started from the sixth scale degree. For instance, If you take the same notes as F major, but instead use D as the tonic note, the resulting scale is D minor.

A minor

E minor

C minor

Harmonic minor scales
You can form a harmonic minor scale from the regular, or 'natural' minor scale by raising seventh scale degree by a semitone. Due to the dissonance of the semitone it allows for stronger harmonic resolution.
I've shown two examples, as the others can easily be created from the natural minor scale by following the same pattern.
D natural minor

A natural minor

Melodic minor scales
Relative to the natural minor, the melodic minor scale raises both the pitch of the sixth and seventh by a semitone, but only where the scale is ascending.
D melodic minor

E melodic minor

Blues scale
The blues scale is formed from the intervals: Minor third, whole tone, semitone, semitone, minor third, whole tone. It is an example of a 6 note, or 'hexatonic' scale.

Pentatonic scales
The term 'Pentatonic scale' refers to any kind of scale that has 5 notes per octave, of which numerous variations exist. These are a selection of the most common ones.
C Major Pentatonic

D Minor Pentatonic

E Minor pentatonic

F Major Pentatonic

D Major Pentatonic

C Minor Pentatonic

Closing notes
I'd recommend choosing one of each type of scale and practising until you instinctively know how to move between the notes. Play them and observe how they sound in relation to each other, and see if you can improvise some music in them.
To practice them, you may find it helpful to break them down into chunks a few notes long, write out the note names, practice each chunk separately until memorised, and finally bring them together.
We also have a scale generator tool to help you practice them, which breaks scales into chunks and provides an audio reference.