  For $\beta<\frac13$, we consider $C^\beta(\mathbb{T}^3\times [0,T])$ weak
solutions of the incompressible Euler equations that do not conserve the
kinetic energy. We prove that for such solutions the closed and non-empty set
of singular times $\mathcal{B}$ satisfies $\dim_{\mathcal{H}}(\mathcal{B})\geq
\frac{2\beta}{1-\beta}$. This lower bound on the Hausdorff dimension of the
singular set in time is intrinsically linked to the H\"older regularity of the
kinetic energy and we conjecture it to be sharp. As a first step in this
direction, for every $\beta<\beta'<\frac{1}{3}$ we are able to construct, via a
convex integration scheme, non-conservative $C^\beta(\mathbb{T}^3\times [0,T])$
weak solutions of the incompressible Euler system such that
$\dim_{\mathcal{H}}(\mathcal{B})\leq
\frac{1}{2}+\frac{1}{2}\frac{2\beta'}{1-\beta'}$. The structure of the wild
solutions that we build allows moreover to deduce non-uniqueness of
$C^\beta(\mathbb{T}^3\times [0,T])$ weak solutions of the Cauchy problem for
Euler from every smooth initial datum.
