  We study isolated quotient singularities by finite and linearly reductive
group schemes (lrq singularities for short) and show that they satisfy many,
but not all, of the known properties of finite quotient singularities in
characteristic zero:
  (1) From the lrq singularity we can recover the group scheme and the quotient
presentation.
  (2) We establish canonical lifts to characteristic zero, which leads to a
bijection between lrq singularities and certain characteristic zero
counterparts.
  (3) We classify subgroup schemes of ${\mathbf{GL}}_d$ and ${\mathbf{SL}}_d$
that correspond to lrq singularities. For $d=2$, this generalises results of
Klein, Brieskorn, and Hashimoto. Also, our classification is closely related to
the spherical space form problem.
  (4) F-regular (resp. F-regular and Gorenstein) surface singularities are
precisely the lrq singularities by finite and linearly reductive subgroup
schemes of ${\mathbf{GL}}_2$ (resp. ${\mathbf{SL}}_2$). This generalises
results of Klein and Du Val.
  (5) Lrq singularities in dimension $\geq 4$ are infinitesimally rigid. We
classify lrq singularities in dimension $3$ that are not infinitesimally rigid
and compute their deformation spaces. This generalises Schlessinger's rigidity
theorem to positive and mixed characteristic.
  Finally, we study Riemenschneider's conjecture in this context, that is,
whether lrq singularities deform to lrq singularities.
