  Consider a host hypergraph $G$ which contains a spanning structure due to
minimum degree considerations. We collect three results proving that if the
edges of $G$ are sampled at the appropriate rate then the spanning structure
still appears with high probability in the sampled hypergraph. We prove such
results for perfect matchings in hypergraphs above Dirac thresholds, for
$K_r$-factors in graphs satisfying the Hajnal--Szemer\'edi minimum degree
condition, and for bounded-degree spanning trees. In each case our proof is
based on constructing a spread measure and then applying recent results on the
(fractional) Kahn--Kalai conjecture connecting the existence of such measures
with probabilistic thresholds.
  For our second result we give a shorter and more general proof of a recent
theorem of Allen, B\"ottcher, Corsten, Davies, Jenssen, Morris, Roberts, and
Skokan which handles the $r=3$ case with different techniques. In particular,
we answer a question of theirs with regards to the number of $K_r$-factors in
graphs satisfying the Hajnal--Szemer\'edi minimum degree condition.
