  The emergent behavior of spin liquids that are born out of geometrical
frustration makes them an intriguing state of matter. We show that in the
quantum kagome antiferromagnet ZnCu$_3$(OH)$_6$SO$_4$ several different
correlated, yet fluctuating states exist. By combining complementary
local-probe techniques with neutron scattering, we discover a crossover from a
critical regime into a gapless spin-liquid phase with decreasing temperature.
An additional unconventional instability of the latter phase leads to a second,
distinct spin-liquid state that is stabilized at the lowest temperatures. We
advance such complex behavior as a feature common to different frustrated
quantum magnets.
