  (Abridged) Water is a key tracer of dynamics and chemistry in low-mass
protostars, but spectrally resolved observations have so far been limited in
sensitivity and angular resolution. In this first systematic survey of
spectrally resolved water emission in low-mass protostellar objects, H2O was
observed in the ground-state transition at 557 GHz with HIFI on Herschel in 29
embedded Class 0 and I protostars. Complementary far-IR and sub-mm continuum
data (including PACS data from our program) are used to constrain the spectral
energy distribution of each source. H2O intensities are compared to inferred
envelope and outflow properties and CO 3-2 emission. H2O emission is detected
in all objects except one. The line profiles are complex and consist of several
kinematic components. The profiles are typically dominated by a broad Gaussian
emission feature, indicating that the bulk of the water emission arises in
outflows, not the quiescent envelope. Several sources show multiple shock
components in either emission or absorption, thus constraining the internal
geometry of the system. Furthermore, the components include inverse P-Cygni
profiles in 7 sources (6 Class 0, 1 Class I) indicative of infalling envelopes,
and regular P-Cygni profiles in 4 sources (3 Class I, 1 Class 0) indicative of
expanding envelopes. "Bullets" moving at >50 km/s are seen in 4 Class 0
sources; 3 of these are new detections. In the outflow, the H2O/CO abundance
ratio as a function of velocity is nearly the same for all sources, increasing
from 10^-3 at <5 km/s to >10^-1 at >10 km/s. The H2O abundance in the outer
envelope is low, ~10^-10. The different H2O profile components show a clear
evolutionary trend: in the Class 0 sources, emission is dominated by outflow
components originating inside an infalling envelope. When the infall diminishes
during the Class I phase, the outflow weakens and H2O emission disappears.
