Symplectic reductions by non-free group actions are, in general, fairly singular topological spaces. But Sjamaar and Lerman (1991) showed that they can nevertheless be decomposed into smooth symplectic manifolds which "fit together nicely", and that the symplectic structures are compatible with a global Poisson bracket on some substitute for the algebra of smooth functions. In this talk, I will explain an extension of Sjamaar–Lerman's result in hyperkähler geometry. I will also present a general Kempf-Ness type theorem which gives explicit complex-algebraic expressions (as GIT quotients) of certain real symplectic reductions of complex affine varieties with non-standard Kähler structures and shifted moment maps. This theorem will then be applied to singular hyperkähler reductions constructed from moduli spaces of solutions to Nahm's equations, giving interesting Lie theoretical examples which can be analyzed algebraically.