  We study singular metrics of constant negative $Q$-curvature in the Euclidean
space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to
the problem \[
  (-\Delta)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash \{0\},
\] under a finite volume condition $\Lambda:=\int_{\mathbb{R}^n}e^{nu}dx$. We
classify all singular solutions of the above equation based on their behavior
at infinity and zero. As a consequence of this, when $n=1,2$, we show that
there is actually no singular solution. Then adapting a variational technique,
we obtain that for any $n\geq 3$ and $\Lambda>0$, the equation admits solutions
with prescribed asymptotic behavior. These solutions correspond to metrics of
constant negative $Q$-curvature, which are either smooth or have a singularity
at the origin of logarithmic or polynomial type. The present paper complements
previous works on the case of positive $Q$-curvature, and also sharpens
previous results in the nonsingular negative $Q$-curvature case.
