  We study linear series on a general curve of genus $g$, whose images are
exceptional with regard to their secant planes. Working in the framework of an
extension of Brill-Noether theory to pairs of linear series, we prove that a
general curve has no linear series with exceptional secant planes, in a very
precise sense, whenever the total number of series is finite. Next, we
partially solve the problem of computing the number of linear series with
exceptional secant planes in a one-parameter family in terms of tautological
classes associated with the family, by evaluating our hypothetical formula
along judiciously-chosen test families. As an application, we compute the
number of linear series with exceptional secant planes on a general curve
equipped with a one-dimensional family of linear series. We pay special
attention to the extremal case of $d$-secant $(d-2)$-planes to
$(2d-1)$-dimensional series, which appears in the study of Hilbert schemes of
points on surfaces. In that case, our formula may be rewritten in terms of
hypergeometric series, which allows us both to prove that it is nonzero and to
deduce its asymptotics in $d$.
