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Sep 3, 2018 at 8:10 comment added Tim Dokchitser Rob de Jeu and Hang Liu have a paper 'On K_2 of certain families of curves', International Mathematics Research Notices, 2015 (2015), issue 21, 10929-10958, arxiv.org/abs/1402.4822. They give examples of the type that you are looking for, I believe.
Sep 3, 2018 at 6:22 comment added David Loeffler Modular curves $X_1(N) / \mathbf{Q}$ are another natural thing to try: the $K_2$ of these curves is rather well understood thanks to Bloch, Beilinson and Kato.
Sep 3, 2018 at 4:58 comment added naf Quite a lot is known about Fermat curves; the paper of Otsubo "On the regulator of Fermat motives and generalized hypergeometric functions" might be a useful place to start.
Sep 3, 2018 at 2:26 history asked user127776 CC BY-SA 4.0