Are they all just 'Em', or is there some standard, concise way of differentiating between them? Thanks!
While the other answers shed light on the issue, they don't answer the question about how to notate different minor scales.
The need for this may arise e.g. when performing harmonic analysis, where one wishes to express the origin of a borrowed chord. In these situations notations such as Em(H), Em(M) and Em(N) can be sometimes used to refer to the harmonic, melodic and natural E minor scales respectively.
Similarly, you may sometimes see notations of borrowed harmonic functions such as IV(M-) to refer to the IV of the (ascending) melodic minor.
However, as the other answers suggest, this differentiation is rarely expressed explicitly. In most cases it is inferred from the actual notes or chords used.
The three examples you give in your question are not different keys, but are different scales. All of these are E minor scales (scales used in the key of E minor). So, a piece using them would have a key signature of one sharp (F#). The differences between them would be notated in the actual music; using C# and D# (instead of C and D) when using E minor melodic ascending; using a D# (instead of D) when using E harmonic minor; E minor melodic descending and E natural minor (also known as E aeolian) use the notes of the key signature for E minor (E F# G A B C D).
Apart from the necessary pitch changes, there is no need to differentiate between these scales when writing a piece of music; they are all E minor and use an E minor key signature. However, if you were describing the scales used in actual passages of a piece in a minor key (for a musical analysis, or to help someone performing the piece, for instance), you may then wish to use the names of different scales.
The key for all 3 is just E minor, relative of G major, so the key signature will always be the F#. In the 3 scales, the first 5 notes are identical - it's only the 6th and 7th notes that vary.The natural minor will have all the same notes as G major; the harmonic minor will have a raised leading note (D#) all the time; the melodic is the odd one, as both 6 and 7 are raised usually on the way up, and are just like the natural minor on the way down.The accidentals, both sharps and the cancelling naturals, are the main clue as to whether the tune is using the melodic minor. I said usually, as the jazz melodic tends to use the raised 6 and 7 most of the time. This sort of makes four minors, and that's before we consider Dorian and maybe Phrygian modes.
So - the notation won't be in the key signature, but in the notes along the way.
A couple of different cases:
1) For natural minor (aeolian), harmonic minor or melodic minor (ascending and descending) the key signature is always the same. In case of E minor, that would be one sharp (at f#). Accidentals are added as needed to denote those pitches that are not by default implied by the key signature. So for melodic minor ascending, you'd use a c# and d# accidentals in the case of E minor.
2) If your piece is consistently in a particular mode, then typically the key signature is in accordance with the relative minor or major key of that mode. That is, the key signature that one would find for the key corresponding to the ionian or aeolian mode.
Example1: if your piece is in E phrygian, you'd have a key signature without any sharps or flats; without analysis, the key signature would imply C major or A minor.
Example2: If your piece is in E dorian, the key signature would be 2 sharps; without analysis the key signature would appear to indicate D major or B minor.
3) If your piece is mostly in a melodic minor or harmonic minor key, with sections that are of the dorian mode of that key's name, then I would probably simply use accidentals to denote the changed notes. So for example, if you piece is in E melodic minor, with some sections in E dorian, then you'd simply use accidentals to denote the c# that is implied by the dorian mode on E.