We present a consistent framework to set limits on properties of light sterile neutrinos coupled to all three active neutrinos using a combination of the latest cosmological data and terrestrial measurements from oscillations, β-decay, and neutrinoless double-β-decay (0νββ) experiments. We directly constrain the full 3+1 active-sterile mixing matrix elements |Uα4|2, with α∈(e,μ,τ), and the mass-squared splitting Δm241≡m24−m21. We find that results for a 3+1 case differ from previously studied 1+1 scenarios where the sterile is coupled to only one of the neutrinos, which is largely explained by parameter space volume effects. Limits on the mass splitting and the mixing matrix elements are currently dominated by the cosmological datasets. The exact results are slightly prior dependent, but we reliably find all matrix elements to be constrained below |Uα4|2≲10−3. Short-baseline neutrino oscillation hints in favor of eV-scale sterile neutrinos are in serious tension with these bounds, irrespective of prior assumptions. We also translate the bounds from the cosmological analysis into constraints on the parameters probed by laboratory searches, such as mβ or mββ, the effective mass parameters probed by β-decay and 0νββ searches, respectively. When allowing for mixing with a light sterile neutrino, cosmology leads to upper bounds of mβ<0.09  eV and mββ<0.07  eV at 95% CL, more stringent than the limits from current laboratory experiments.