We discuss the linear and two-photon spectroscopic selection rules for spin-singlet excitons in monolayer transition-metal dichalcogenides. Our microscopic formalism combines a fully k-dependent few-orbital band structure with a many-body Bethe-Salpeter equation treatment of the electron-hole interaction, using a model dielectric function. We show analytically and numerically that the single-particle, valley-dependent selection rules are preserved in the presence of excitonic effects. Furthermore, we definitively demonstrate that the bright (one-photon allowed) excitons have s-type azimuthal symmetry and that dark p-type excitons can be probed via two-photon spectroscopy. The screened Coulomb interaction in these materials substantially deviates from the 1/ epsilon sub(0)r form; this breaks the "accidental" angular momentum degeneracy in the exciton spectrum, such that the exciton has a lower energy than the exciton by at least 50 meV. We compare our calculated two-photon absorption spectra to recent experimental measurements.