In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.