Let's start with an excellent performance by Argerich, Kremer and Maisky:
This is a fairly short movement, just over three minutes. Just because scherzos are quick doesn't mean they have to be brief--there are lots of examples by Beethoven that are twice as long as this. Where this music departs significantly from the Beethoven model is that while 18th and 19th century scherzos had a great deal of repetition, this one does not. Themes are repeated, of course, but they are being continually transformed. Here is that opening theme in the violin:
Very classical in its adherence to outlining triads. The opening leap of a fourth followed by a fifth is probably a bit inelegant for Haydn or even Beethoven, but the rest is rather traditional in character. The piano accompanies this with harmonies that quickly wander from traditional norms, however. The opening tonic is followed by a Neapolitan, then a G# major harmony! Rhythmically the typical kinds of patterns one would find in a 3/4 scherzo all seem just a bit "off" as witnessed by this maniacal little passage for the cello. The frenetic character is only heightened by its being in a very high register:
Instead of a contrasting trio in perhaps the dominant as would be found in a Beethoven scherzo, we have a waltz-like section in G major, the key of the Neapolitan. Here is how it starts:
And the continuation with a theme in the cello:
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A couple of things to notice about the style: the accompaniment is quite conventional, but the the cello manages to wander into G minor briefly, which gives an uneasy feeling to the passage. Shostakovich had a particular gift for both using traditional forms and harmonies, but at the same time subverting them in subtle and unsubtle ways.
After this contrasting section, the piece returns to the key and themes of the beginning, but in varied form. For example, the repeat of the opening begins with the same music, but this time the theme is in the piano, accompanied by the strings:
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(It is an exact repeat, I just left off the upbeat melody note because it was in the previous line.)
Another thing to notice about the movement is how terribly Russian it is in its wild, heavy, and slightly drunken mood. This is achieved by a hundred little details: the sometimes clumsy phrases (intentionally so, of course) that just seem to land slightly off-balance, the odd spacings, the strange twists of the melody and the woozy glissandi.
Let's listen to the movement again. Here is a slightly brisker performance by the Mosa Trio from the Netherlands: