Mnemonics, up and down
Going up: Jaws. Two notes a semitone apart, over and over, faster and faster; half a century later it's still the most famous minor 2nd in the world. The Pink Panther theme sneaks up by the same interval into its first long note.
Going down: Für Elise. The opening E to D♯ shimmer is a falling minor 2nd repeated, and Joy to the World starts its long descent home with one. The seconds are gentler going down: less menace, more sigh. Both directions are a button away above.
Jazz keeps its own reference tunes. Mark Levine's interval table fixes the ascending minor 2nd to Monk's "Blue Monk" and the descending one to Cedar Walton's "Bolivia."
On the staff
The smallest distance, the strongest pull
Both seconds are dissonances, and the minor 2nd is the sharper of the pair. Sound the two notes together and they grind; the interval barely fits in its own skin. But its real career is melodic. The semitone from ti up to do is the leading tone resolving home, the little gravity well at the end of nearly every cadence in tonal music. Train your ear on this interval and you're really training it on the feeling of almost-there.
One spelling quirk worth knowing: notation distinguishes C to D♭ (a minor 2nd, two letter names) from C to C♯ (an augmented unison, one letter), even though both span a single semitone. The ear hears one interval; the page writes two. Which spelling a composer picks tells you where the line is headed, not how it sounds.
What it gets confused with
The major 2nd, one semitone wider, is the discrimination that matters. Both are steps, but the major 2nd is the ordinary walking step of scales, while the minor 2nd is cramped, urgent, a foot caught mid-stride. If the two notes together sound like an everyday passing moment, it's probably the major; if they bristle, the minor. Heard melodically, the minor 2nd barely seems to move at all, which is itself the tell.
How Aubel handles it
The minor 2nd unlocks sixth beyond the starting set, immediately after its sibling the minor 3rd. By then the major 2nd has been in the pool long enough to serve as the reference step, and the new interval's job is to sound one size too small. The detail screen leads with Jaws, because some mnemonics never let go.