  Classical Cepheids provide valuable insights into the evolution of stellar
multiplicity among intermediate-mass stars. Here, we present a systematic
investigation of single-lined spectroscopic binaries (SB1) based on
high-precision velocities measured by the VELOcities of CEpheids (VELOCE)
project. We detected 76 (29%) SB1 systems among the 258 Milky Way Cepheids in
the first VELOCE data release, 32 (43%) of which were not previously known to
be SB1 systems. We determined 30 precise and 3 tentative orbital solutions, 18
(53%) of which are reported for the first time. This large set of Cepheid
orbits provides a detailed view of the eccentricity e and orbital period Porb
distribution among evolved intermediate-mass stars, ranging from e=[0.0, 0.8]
and Porb=[240, 9 000] d. Orbital motion on timescales exceeding the 11 yr
VELOCE baseline was investigated using a template fitting technique applied to
literature data. Particularly interesting objects include a) R Cru, the Cepheid
with the shortest orbital period in the Milky Way (240 d), b) ASAS
J103158-5814.7, a short-period overtone Cepheid exhibiting time-dependent
pulsation amplitudes as well as orbital motion, c) 17 triple systems with outer
visual companions, among other interesting objects. Most VELOCE Cepheids
(21/23) that exhibit evidence for a companion based on Gaia proper motion
anomaly are also spectroscopic binaries, whereas the remaining do not exhibit
significant (> 3-sigma) orbital RV variations. Gaia quality flags, notably the
Renormalized Unit Weight Error (RUWE), do not allow to reliably identify
Cepheid binaries, although statistically the average RUWE of SB1 Cepheids is
slightly higher than that of non-SB1 Cepheids. Comparison with Gaia photometric
amplitudes in G, Bp, and Rp also does not allow to identify spectroscopic
binaries among the full VELOCE sample.
