I will describe briefly constructions of families of projective quotients of Hilbert schemes of curves and their applications both as log-canonical models of the moduli space of stable curves and as moduli spaces in their own right for new classes of curves. Because a degree parameter that could be taken sufficiently large in prior constructions must be fixed to go further, classical asymptotic methods for analysing stability are no longer effective. After reviewing this setup, I will outline a new method, joint with David Swinarski, for analysing stability in fixed degree for very special varieties, and conclude by explaining how the method is applied in recent work of Jarod Alper, Maksym Fedorchuk and David Smyth.