  We study certain families of oscillatory integrals $I_\varphi(a)$,
parametrised by phase functions $\varphi$ and amplitude functions $a$ globally
defined on $\mathbb{R}^d$, which give rise to tempered distributions, avoiding
the standard homogeneity requirement on the phase function. The singularities
of $I_\varphi(a)$ are described both from the point of view of the lack of
smoothness as well as with respect to the decay at infinity. In particular, the
latter will depend on a version of the set of stationary points of $\varphi$,
including elements lying at the boundary of the radial compactification of
$\mathbb{R}^d$. As applications, we consider some properties of the two-point
function of a free, massive, scalar relativistic field and of classes of global
Fourier integral operators on $\mathbb{R}^d$, with the latter defined in terms
of kernels of the form $I_\varphi(a)$.
