We construct radial and non-radial singular solutions u ∈ C4(B∖{0}) with a non-removable singular point x = 0 for the conformal Q-curvature equation \(\begin{cases}\Delta^{2}u={\rm{e}}^{u}\quad\text{in}\,B\backslash\{0\},\\\int_{B\backslash\{0\}}{\rm{e}}^{u(x)}\,dx<\infty,\end{cases}\) where B = {x ∈ ℝ4: ∣x∣ < 1}. More precisely, we can construct two different types of singular solutions u ∈ C4(B∖{0}) of the equation, satisfying \(\vert x\vert^{2}u(x)\rightarrow - D<0\quad\text{uniformly}\,\text{as}\,\vert x\vert \rightarrow 0\) for some D0 ≥ 0 and any D > D0 ≥ 0, and satisfying \(u(x)=o(\vert x\vert^{-2})\quad\text{uniformly}\,\text{as}\,\vert x\vert \rightarrow 0\) Moreover, detailed asymptotic expansions near x = 0 of these radial and non-radial singular solutions can be established. As an application, we can also obtain the existence of two types of solutions \(u\in C^{4}(\mathbb{R}^{4}\backslash\overline{B})\) to the problem \(\begin{cases}\Delta^{2}u={\rm{e}}^{u}\quad\text{in}\,\mathbb{R}^{4}\backslash\overline{B},\\\int_{\mathbb{R}^{4}\backslash\overline{B}}{\rm{e}}^{u(x)} dx<\infty\end{cases}\) satisfying |x|−2u(x) → − D < 0 uniformly as |x| → ∞ for some D0 ≥ 0 and any D > D0, and satisfying u(x) = o(|x|2) uniformly as |x| → ∞.
