  The fine structure of the exciton spectrum, containing optically allowed
(bright) and forbidden (dark) exciton states, determines the radiation
efficiency in nanostructures. We study time-resolved micro-photoluminescence in
MoS$_2$ monolayers and bilayers, both unstrained and compressively strained, in
a wide temperature range (10-300 K) to distinguish between exciton states
optically allowed and forbidden, both in spin and momentum, as well as to
estimate their characteristic decay times and contributions to the total
radiation intensity. The decay times were found to either increase or decrease
with increasing temperature, indicating the lowest bright or lowest dark state,
respectively. Our results unambiguously show that, in an unstrained monolayer,
the spin-allowed state is the lowest for a series of A excitons (1.9 eV) with
the dark state being < 2 meV higher, and that the splitting energy can increase
several times at compression. In contrast, in the indirect exciton series in
bilayers (1.5 eV), the spin-forbidden state is the lowest, being about 3 meV
below the bright one. The strong effect of strain on the exciton spectrum can
explain the large scatter among the published data and must be taken into
account to realize the desired optical properties of 2D MoS$_2$.
