We present the need for new fractional spectral theories, explicitly yielding rather non-polynomial, yet orthogonal, eigensolutions to effectively represent the singularities in solutions to FODEs/FPDEs. To this end, we present the regular/singular theories of fractional Sturm–Liouville eigen-problems. We call the corresponding explicit eigenfunctions of these problems Jacobi poly-fractonomials. We demonstrate their attractive properties including their analytic fractional derivatives/integrals, three-term recursions, special values, function approximability, etc. Subsequently, we introduce the notion of generalized Jacobi poly-fractonomials (GJPFs), expanding the range of admissible parameters also allowing function singularities of negative indices at both ends. Next, we present a rigorous approximation theory for GJPFs with numerical examples. We further generalize our fractional Sturm–Liouville theories to regular/singular tempered fractional Sturm–Liouville eigen-problems, where a new exponentially tempered family of fractional orthogonal basis functions emerges. We finally introduce a variant of orthogonal basis functions suitable for anomalous transport that occurs over significantly longer time-periods.