  We analyze the expected arrival time spectrum of supernova neutrinos using
simulated luminosity and compute the expected number of events in future
detectors such as the DUNE Far Detector and Hyper-Kamiokande. We develop a
general method using minimum square statistics that can compute the sensitivity
to any variable affecting neutrino time of flight. We apply this method in two
different situations: First, we compare the time spectrum changes due to
different neutrino mass values to put limits on electron (anti)neutrino
effective mass. Second, we constrain Lorentz invariance violation through the
mass scale, $M_{QG}$, at which it would occur. We consider two main neutrino
detection techniques: 1. DUNE-like liquid argon TPC, for which the main
detection channel is $\nu_e +\, ^{40}\mbox{Ar} \rightarrow e^- +\,
^{40}\mbox{K}^*$, related to the supernova neutronization burst; and 2.
HyperK-like water Cherenkov detector, for which $\bar \nu_e + p \rightarrow e^+
+ n$ is the main detection channel. We consider a fixed supernova distance of
10~kpc and two different masses of the progenitor star: (i) 15~$M_\odot$ with
neutrino emission time up to 0.3~s and (ii) 11.2~$M_\odot$ with neutrino
emission time up to 10~s. The best mass limits at 3$\sigma$ are for
$\mathcal{O}(1)$~eV. For $\nu_e$, the best limit comes from a DUNE-like
detector if the mass ordering happens to be inverted. For $\bar \nu_e$, the
best limit comes from a HyperK-like detector. The best limit for the Lorentz
invariance violation mass scale at the 3$\sigma$ level considering a
superluminal or subluminal effect is $M_{QG} \gtrsim 10^{13}$~GeV ($M_{QG}
\gtrsim 5 \times 10^{5}$~GeV) for linear (quadratic) energy dependence.
