In single player, banners will sway back and forth slightly. In Multiplayer, they do not.
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But it looks like it doesn't happen in LAN world

On my SMP server they are swaying
This actually seems to be an age-related issue - should I create a new issue with a more accurate title/description?

What do you mean, age related?
When I created a new SSP world, the banners swayed as expected. When I hopped on my several-month-old server, they did not. When I copied the DayTime and Time NBT tags from the server's level.dat to the SSP world's level.dat, the banners did not sway in that world. I'm guessing it's due to larger double-precision numbers aren't as accurate after the decimal.

Please make a new ticket for that. Comment here with a link to the ticket and I can make sure its all linked up and referenced.

Reopened.
Can confirm in SMP.

... but duplicate of MC-63720 for "old" worlds
Well to be fair, this was reported a day before MC-63720, but if that one has better info, then it's fine
It happens to me too.