We search for the presence of bright Lyα emitters among Spitzer SMUVS galaxies at z > 2.9 making use of homogeneous MUSE spectroscopic data. Although these data only cover a small region of COSMOS, MUSE has the unique advantage of providing spectral information over the entire field, without the need of target pre-selection. This results in an unbiased detection of all the brightest Lyα emitters among the SMUVS sources, which by design are stellar-mass selected galaxies. Within the studied area, ∼14% of the SMUVS galaxies at z > 2.9 have Lyα fluxes Fλ ≳ 7 × 10-18 erg s-1 cm-2. These Lyα emitters are characterized by three types of emission, 47% show a single-line profile, 19% present a double peak or a blue bump, and 31% show a red tail. One object (3%) shows both a blue bump and a red tail. We also investigate the spectral energy distribution (SED) properties of the SMUVS galaxies that are MUSE detected and those that are not. After stellar mass matching the two populations, we find that the MUSE detected galaxies have generally lower extinction than SMUVS-only objects, while there is no clear intrinsic difference in the mass and age distributions of the two samples. For the MUSE-detected SMUVS galaxies, we compare the instantaneous star formation rate lower limit obtained from the Lyα line with its past average derived from SED fitting, and find evidence for rejuvenation in some of our oldest objects. In addition, we study the spectra of those Lyα emitters that are not detected in SMUVS in the same field. We find that of the emission line profiles shown 67% have a single line, 3% a blue bump, and 30% a red tail. The difference in profile distribution could be ascribed to the fainter Lyα luminosities of the MUSE sources not detected in SMUVS and an intrinsically different mass distribution. Finally, we search for the presence of galaxy associations using the spectral redshifts. The integral coverage of MUSE reveals that these associations are 20 times more likely than what is derived from all the other existing spectral data in COSMOS, which is biased by target pre-selection.