  Several state-of-the-art numerical methods have observed static or
fluctuating spin and charge stripes in doped two-dimensional Hubbard models,
suggesting that these orders play a significant role in shaping the cuprate
phase diagram. Many experiments, however, also indicate that the cuprates have
strong electron-phonon ($e$-ph) coupling, and it is unclear how this
interaction influences stripe correlations. We study static and fluctuating
stripe orders in the doped singleband Hubbard-Holstein model using zero
temperature variational Monte Carlo and finite temperature determinant quantum
Monte Carlo. We find that the lattice couples more strongly with the charge
component of the stripes, leading to an enhancement or suppression of stripe
correlations, depending on model parameters like the next-nearest-neighbor
hopping $t^\prime$ or phonon energy $\Omega$. Our results help elucidate how
the $e$-ph interaction can tip the delicate balance between stripe and
superconducting correlations in the Hubbard-Holstein model with implications
for our understanding of the high-$T_\mathrm{c}$ cuprates.
