  We present measurements of the specific star-formation rate (SSFR)-stellar
mass relation for star-forming galaxies. Our deep spectroscopic samples are
based on the Redshift One LDSS3 Emission line Survey, ROLES, and European
Southern Observatory, ESO, public spectroscopy at z=1, and on the Sloan Digital
Sky Survey (SDSS) at z=0.1. These datasets cover an equally deep mass range of
8.5<~log(M*/Msun)<~11 at both epochs. We find that the SSFR--mass relation
evolves in a way which is remarkably independent of stellar mass, as we
previously found for the star-formation rate density (SFRD)--mass relation. At
higher masses, such as those probed by previous surveys, the evolution in
SSFR--mass is almost independent of stellar mass. At higher masses
(log(M*/Msun)>10) the shapes of the cumulative cosmic SFRDs are very similar at
both z=0.1 and z=1.0, both showing 70% of the total SFRD above a mass of
log(M*/Msun)>10. Mass functions are constructed for star-forming galaxies and
found to evolve by only <35% between z=1 and z=0.1 over the whole mass range.
The evolution is such that the mass function decreases with increasing cosmic
time, confirming that galaxies are leaving the star-forming sequence/blue
cloud. The observational results are extended to z~2 by adding two recent Lyman
break galaxy samples, and data at these three epochs (z=0.1, 1, 2) are compared
with the GALFORM semi-analytic model of galaxy formation. GALFORM predicts an
overall SFR density (SFRD) as a function of stellar mass in reasonable
agreement with the observations. The star formation timescales inferred from
1/SSFR also give reasonable overall agreement, with the agreement becoming
worse at the lowest and highest masses. [abridged]
