  We consider the parabolic-elliptic Keller-Segel system in spatial dimensions
$d\geq3$, which corresponds to the mass supercritical case. Some solutions
become singular in finite time, an important example being backward
self-similar solutions. Herrero et al. and Brenner et al. showed the existence
of such profiles, countably many in dimensions $3\leq d \leq 9$ and at least
two for $d\geq 10$. We establish that all these self-similar profiles are
stable along a set of initial data with finite Lipschitz codimension equal to
the number of instable eigenmodes. This extends the recent finding of Glogi\'c
et al. showing the stability of the fundamental self-similar profile. We obtain
additional results, such as the possibility of the solutions we construct to
originate from smooth and compactly supported initial data, their convergence
at blow-up time, and the Lipschitz regularity of the blow-up time. Our proof
extends the approach proposed in Collot et al., based on renormalizing the
solution around a modulated self-similar solution, and using a spectral gap for
the linearized operator in the parabolic neighbourhood of the singularity.
