  We perform a comprehensive study of cosmological constraints on non-standard
neutrino self-interactions using cosmic microwave background (CMB) and baryon
acoustic oscillation data. We consider different scenarios for neutrino
self-interactions distinguished by the fraction of neutrino states allowed to
participate in self-interactions and how the relativistic energy density,
N$_{\textrm{eff}}$, is allowed to vary. Specifically, we study cases in which:
all neutrino states self-interact and N$_{\textrm{eff}}$ varies; two species
free-stream, which we show alleviates tension with laboratory constraints,
while the energy in the additional interacting states varies; and a variable
fraction of neutrinos self-interact with either the total N$_{\textrm{eff}}$
fixed to the Standard Model value or allowed to vary. In no case do we find
compelling evidence for new neutrino interactions or non-standard values of
N$_{\textrm{eff}}$. In several cases we find additional modes with neutrino
decoupling occurring at lower redshifts $z_{\textrm{dec}} \sim 10^{3-4}$. We do
a careful analysis to examine whether new neutrino self-interactions solve or
alleviate the so-called $H_0$ tension and find that, when all Planck 2018 CMB
temperature and polarization data is included, none of these examples ease the
tension more than allowing a variable N$_{\textrm{eff}}$ comprised of
free-streaming particles. Although we focus on neutrino interactions, these
constraints are applicable to any light relic particle.
