  In this paper, we study the existence of positive functions $K \in
C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation
\begin{equation}\label{001} P_m (v) =K v^{\frac{n+2m}{n-2m}}~~~~~~ {on} ~
\mathbb{S}^n \{equation} has a singular positive solution $v$ whose singular
set is a single point, where $m$ is an integer satisfying $1 \leq m < n/2$ and
$P_m$ is the intertwining operator of order $2m$. More specifically, we show
that when $n\geq 2m+4$, every positive function in $C^1(\mathbb{S}^n)$ can be
approximated in the $C^1(\mathbb{S}^n)$ norm by a positive function $K\in
C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation has a
singular positive solution whose singular set is a single point. Moreover, such
a solution can be constructed to be arbitrarily large near its singularity.
This is in contrast to the well-known results of Lin \cite{Lin1998} and Wei-Xu
\cite{Wei1999} which show that the conformal $Q$-curvature equation, with $K$
identically a positive constant on $\mathbb{S}^n$, $n > 2m$, does not exist a
singular positive solution whose singular set is a single point.
