  We define and study the vanishing sequence along a real valuation of sections
of a line bundle on a projective variety. Building on previous work of the
first author with Huayi Chen, we prove an equidistribution result for vanishing
sequences of large powers of a big line bundle, and study the limit measure. In
particular, the latter is described in terms of restricted volumes for
divisorial valuations. We also show on an example that the associated concave
function on the Okounkov body can be discontinuous at boundary points.
