  For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we
obtain $k$-explicit analogues of the classic local FEM error bounds of
[Nitsche, Schatz 1974], [Wahlbin 1991], [Demlow, Guzm\'an, Schatz 2011],
showing that these bounds hold with constants independent of $k$, provided one
works in Sobolev norms weighted with $k$ in the natural way.
  We prove two main results: (i) a bound on the local $H^1$ error by the best
approximation error plus the $L^2$ error, both on a slightly larger set, and
(ii) the bound in (i) but now with the $L^2$ error replaced by the error in a
negative Sobolev norm. The result (i) is valid for shape-regular
triangulations, and is the $k$-explicit analogue of the main result of [Demlow,
Guzm\'an, Schatz, 2011]. The result (ii) is valid when the mesh is locally
quasi-uniform on the scale of the wavelength (i.e., on the scale of $k^{-1}$)
and is the $k$-explicit analogue of the results of [Nitsche, Schatz 1974],
[Wahlbin 1991].
  Since our Sobolev spaces are weighted with $k$ in the natural way, the result
(ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo
low frequencies (i.e., frequencies $\lesssim k$). Numerical experiments confirm
this property, and also highlight interesting propagation phenomena in the
Helmholtz FEM error.
