  We perform a stringent search for precursor emission of short gamma-ray
bursts (SGRBs) from the Fermi/GBM data and find 16 precursor events with
$\gtrsim4.5\sigma$ significance. We find that the durations of the main SGRB
emission ($T_{\rm GRB}$) and the precursor emission ($T_{\rm pre}$), as well as
the waiting time ($T_{\rm wt}$) in between, are roughly comparable to each
other, with $T_{\rm wt}\approx2.8T_{\rm GRB}^{1.2}$ approximately satisfied for
most cases except one significant outlier. We also perform spectral analyses to
the precursors and SGRBs, and find that the spectra of precursor emission can
be fitted with the blackbody, non-thermal cutoff power law and/or power law
models. We consider several possible models for precursor emission in SGRBs and
find that the luminosity and spectral shape may be explained by the the shock
breakout or the photospheric radiation of a fireball launched after the merger
for thermal precursors, or magnetospheric interaction between two NSs prior to
the merger for non-thermal precursors. For the fireball photospheric model, a
matter-dominated jet is preferred and a constraint on the fireball Lorentz
factor can be placed as $\Gamma\sim30$. For the magnetospheric interaction
model, jet launching mechanism may be constrained. In particular, those events
with $T_{\rm wt}/T_{\rm GRB}\gg1$ (e.g. GRB191221802) require the formation of
a supramassive or stable neutron star after the merger, with the delay time
defined by the timescale for an initially baryon-loaded jet to become
magnetically dominated and relativistic.
