  In this paper, we introduce a new family of orthogonal systems, termed as the
M\"{u}ntz ball polynomials (MBPs), which are orthogonal with respect to the
weight function: $\|x\|^{2\theta+2\mu-2} (1-\|x\|^{2\theta})^{\alpha}$ with the
parameters $\alpha>-1, \mu>- 1/2$ and $\theta>0$ in the $d$-dimensional unit
ball $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$. We then
develop efficient and spectrally accurate MBP spectral-Galerkin methods for
singular eigenvalue problems including degenerating elliptic problems with
perturbed ellipticity and Schr\"odinger's operators with fractional potentials.
We demonstrate that the use of such non-standard basis functions can not only
tailor to the singularity of the solutions but also lead to sparse linear
systems which can be solved efficiently.
