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Sevenths always resolve down. The resolution may be delayed by expansions and embellishments, but eventually the seventh will still resolve.
Sevenths are usually prepared before resolution. This means that it is either treated like a suspension or approached by step. In predominant 7th chords (ii7, iiø7, iv7) the seventh is usually treated like a suspension. This means it begins as a chord tone in the previous chord, becomes the seventh obliquely (stays on the same note), and then resolves down in the following chord.
Example of seventh preparation:
^8 (I) – ^8(ii7) – ^7(V)
Tritones, which occur in dominant 7ths and diminished/half-diminished chords, must resolve in contrary motion: x4 outwards, o5 inwards. If in doubt, look for the leading tone, which always resolves up, and the 7th, which always resolves down.
Example of tritone resolution:
^4 – ^3
^7 – ^8
V7 – I
The 3rd of the diminished 7th has an exception: it can resolve down so that the root, not the third, is doubled in the next chord.
Example of alternate resolution:
^6 – ^5
^4 – ^3
^2 – ^1 (resolves down instead of up to ^3)
^7 – ^8
viio7 – i
In some cases, if scale degree ^4 (the 7th of V7) is in soprano it can resolve up, if and only if this forms parallel 10ths with another rising voice. Depending on chord voicing, this can form parallel 5ths which may be called “Mozart 5ths”.
Example of parallel 10ths:
^3 – ^4 – ^5 (resolves up, not down)
^8 – ^7 – ^8
^5 – ^5 – ^5
^1 – ^2 – ^3 (forms parallel 10ths)
I – V43 – I6