Fourier transform of a periodic/aperiodic signal
Equidistant Dirac impulses in the spectrum imply a periodic time domain signal.

As pointed out in a comment, in continuous time, the signal $x(t)=\sin(\omega_0t)$ is always periodic, regardless of the value of $\omega_0$.

Your question about the spectrum being periodic is unclear to me.

A one-sided spectrum with equidistant Dirac impulses also implies a periodic time-domain signal, but that signal is necessarily complex-valued because its spectrum has no Hermitian symmetry, as required for real-valued signals.