  The times of Brownian local minima, maxima and their union are three distinct
examples of local, stationary, dense, random countable sets associated with
classical Wiener noise. Being local means, roughly, determined by the local
behavior of the sample paths of the Brownian motion, and stationary means
invariant relative to the L\'evy shifts of the sample paths. We answer to the
affirmative Tsirelson's question, whether or not there are any others, and
develop some general theory for such sets. An extra ingredient to their
structure, that of an honest indexation, leads to a splitting result that is
akin to the Wiener-Hopf factorization of the Brownian motion at the minimum (or
maximum) and has the latter as a special case. Sets admitting an honest
indexation are moreover shown to have the property that no stopping time
belongs to them with positive probability. They are also minimal: they do not
have any non-empty proper local stationary subsets. Random sets, of the kind
studied in this paper, honestly indexed or otherwise, give rise to nonclassical
one-dimensional noises, generalizing the noise of splitting. Some properties of
these noises and the inter-relations between them are investigated. In
particular, subsets are connected to subnoises.
