  Let $(M,\textit{g},\sigma)$ be an $m$-dimensional closed spin manifold, with
a fixed Riemannian metric $\textit{g}$ and a fixed spin structure $\sigma$; let
$\mathbb{S}(M)$ be the spinor bundle over $M$. The spinorial Yamabe-type
problems address the solvability of the following equation \[
D_{\textit{g}}\psi = f(x)|\psi|_{\textit{g}}^{\frac2{m-1}}\psi, \quad
\psi:M\to\mathbb{S}(M), \ x\in M \] where $D_{\textit{g}}$ is the associated
Dirac operator and $f:M\to\mathbb{R}$ is a given function. The study of such
nonlinear equation is motivated by its important applications in Spin Geometry:
when $m=2$, a solution corresponds to a conformal isometric immersion of the
universal covering $\widetilde M$ into $\mathbb{R}^3$ with prescribed mean
curvature $f$; meanwhile, for general dimensions and $f\equiv constant\neq0$, a
solution provides an upper bound estimate for the B\"ar-Hijazi-Lott invariant.
  The aim of this paper is to establish non-compactness results related to the
spinorial Yamabe-type problems. Precisely, concrete analysis is made for two
specific models on the manifold $(S^m,\textit{g})$ where the solution set of
the spinorial Yamabe-type problem is not compact: $1).$ the geometric potential
$f$ is constant (say $f\equiv1$) with the background metric $\textit{g}$ being
a $C^k$ perturbation of the canonical round metric $\textit{g}_{S^m}$, which is
not conformally flat somewhere on $S^m$; $2).$ $f$ is a perturbation from
constant and is of class $C^2$, while the background metric
$\textit{g}\equiv\textit{g}_{S^m}$.
