  A splinter is a notion of singularity that has seen numerous recent
applications, especially in connection with the direct summand theorem, the
mixed characteristic minimal model program, Cohen-Macaulayness of absolute
integral closures and cohomology vanishing theorems. Nevertheless, many basic
questions about these singularities remain elusive. One outstanding problem is
whether the splinter property spreads from a point to an open neighborhood of a
noetherian scheme. Our paper addresses this problem in prime characteristic,
where we show that a locally noetherian scheme that has finite Frobenius or
that is locally essentially of finite type over a quasi-excellent local ring
has an open splinter locus. In particular, all varieties over fields of
positive characteristic have open splinter loci. Intimate connections are
established between the openness of splinter loci and $F$-compatible ideals,
which are prime characteristic analogues of log canonical centers. We prove the
surprising fact that for a large class of noetherian rings with pure (aka
universally injective) Frobenius, the splinter condition is detected by the
splitting of a single generically \'etale finite extension. We also show that
for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal
ideal detects the splinter property.
