  Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth
positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta
u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$,
$2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $\mu>
0$. When $\mu=0$ and $s=0$, the profile at the singularity $0$ was fully
described by Caffarelli-Gidas-Spruck. We prove that when $\mu>0$ and $s>0$,
besides this profile, two new profiles might occur. We provide a full
description of all the singular profiles. Special attention is accorded to
solutions such that $\liminf_{x\to 0}|x|^{\frac{n-2}{2}}u(x)=0$ and
$\limsup_{x\to 0}|x|^{\frac{n-2}{2}}u(x)\in (0,+\infty)$. The particular case
$q=(n+2)/(n-2)$ requires a separate analysis which we also perform.
