Context. New laboratory investigations of the rotational spectrum of postulated astronomical species are essential to support the assignment and analysis of current astronomical surveys. In particular, considerable interest surrounds sulfur analogs of oxygen-containing interstellar molecules and their isomers. Aims. To enable reliable interstellar searches of vinyl mercaptan, the sulfur-containing analog to the astronomical species vinyl alcohol, we investigated its pure rotational spectrum at millimeter wavelengths. Methods. We extended the pure rotational investigation of the two isomers syn and anti vinyl mercaptan to the millimeter domain using a frequency-multiplication spectrometer. The species were produced by a radiofrequency discharge in 1,2-ethanedithiol. Additional transitions were remeasured in the centimeter band using Fourier-transform microwave spectroscopy to better determine rest frequencies of transitions with low-J and low-K_a values. Experimental investigations were supported by quantum chemical calculations on the energetics of both the [C₂,H₄,S] and [C₂,H₄,O] isomeric families. Interstellar searches for both syn and anti vinyl mercaptan as well as vinyl alcohol were performed in the EMoCA spectral line survey carried out toward Sgr B2(N2) with ALMA. Results. Highly accurate experimental frequencies (to better than 100 kHz accuracy) for both syn and anti isomers of vinyl mercaptan are measured up to 250 GHz; these deviate considerably from predictions based on extrapolation of previous microwave measurements. Reliable frequency predictions of the astronomically most interesting millimeter-wave lines for these two species can now be derived from the best-fit spectroscopic constants. From the energetic investigations, the four lowest singlet isomers of the [C₂,H₄,S] family are calculated to be nearly isoenergetic, which makes this family a fairly unique test bed for assessing possible reaction pathways. Upper limits for the column density of syn and anti vinyl mercaptan are derived toward the extremely molecule-rich star-forming region Sgr B2(N2) enabling comparison with selected complex organic molecules.