  We prove local measure bounds on the tubular neighbourhood of the singular
set of codimension one stationary integral $n$-varifolds $V$ in Riemannian
manifolds which have both: (i) finite index on their smoothly embedded part;
and (ii) $\mathcal{H}^{n-1}$-null singular set. A direct consequence of such a
bound is a bound on the upper Minkowski content of the singular set of such a
varifold in terms of its total mass and its index. Such a result improves on
known bounds, namely the corresponding bound on the $\mathcal{H}^{n-7}$-measure
of the singular set established by A. Song ([10]), as well as the same bounds
established by A. Naber and D. Valtorta ([6]) for codimension one area
minimising currents. Our results also provide more structural information on
the singular set for codimension one integral varifolds with finite index (on
the regular part) and no classical singularities established by N.
Wickramasekera ([11]).
