  We prove a slope 1 stability range for the homology of the symplectic,
orthogonal and unitary groups with respect to the hyperbolic form, over any
fields other than $F_2$, improving the known range by a factor 2 in the case of
finite fields. Our result more generally applies to the automorphism groups of
vector spaces equipped with a possibly degenerate form (in the sense of Bak,
Tits and Wall). For finite fields of odd characteristic, and more generally
fields in which -1 is a sum of two squares, we deduce a stability range for the
orthogonal groups with respect to the Euclidean form, and a corresponding
result for the unitary groups.
  In addition, we include an exposition of Quillen's unpublished slope 1
stability argument for the general linear groups over fields other than $F_2$,
and use it to recover also the improved range of
Galatius-Kupers-Randal-Williams in the case of finite fields, at the
characteristic.
