  We show that in two-band $s$-wave superfluids it is possible to induce
quantum phase transitions (QPTs) by tuning intraband and interband $s$-wave
interactions, in sharp contrast to single-band $s$-wave superfluids, where only
a crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein
condensation (BEC) superfluidity occurs. For non-zero interband and attractive
intraband interactions, we demonstrate that the ground state has always two
interpenetrating superfluids possessing three spectroscopically distinct
regions where pairing is qualitatively different: (I) BCS pairing in both bands
(BCS-BCS), (II) BCS pairing in one band and BEC pairing in the other (BCS-BEC),
and (III) Bose pairing in both bands (BEC-BEC). Furthermore, we show that by
fine tuning the interband interactions to zero one can induce QPTs in the
ground state between three distinct superfluid phases. There are two phases
where only one band is superfluid ($S_1$ or $S_2$), and one phase where both
bands are superfluid $(S_1 + S_2)$, a situation which is absent in one-band
$s$-wave systems. Lastly, we suggest that these crossovers and QPTs may be
observed in $^{173}$Yb and $^{87}$Sr.
