  We consider the reaction-diffusion equation
  \begin{equation*}
  u_t=u_{xx}+\mu\left(\int_\Omega M(y,z)u(t,x,z)dz-u\right) +
u\left(a(y)-\int_\Omega K(y,z) u(t,x,z)dz\right) ,
  \end{equation*}
  where $ u=u(t,x,y) $ stands for the density of a theoretical population with
a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$)
structure, $ M(y,z) $ is a mutation kernel acting on the phenotypic space, $
a(y) $ is a fitness function and $ K(y,z) $ is a competition kernel. Using a
vanishing viscosity method, we construct measure-valued traveling waves for
this equation, and present particular cases where singular traveling waves do
exist. We determine that the speed of the constructed traveling waves is the
expected spreading speed $ c^*:=2\sqrt{-\lambda_1} $, where $ \lambda_1 $ is
the principal eigenvalue of the linearized equation. As far as we know, this is
the first construction of a measure-valued traveling wave for a
reaction-diffusion equation.
