  A detection of primordial non-Gaussianity could transform our understanding
of the fundamental theory of inflation. The precision promised by upcoming CMB
and large-scale structure surveys raises a natural question: if a detection
given a particular template is made, what does this truly tell us about the
underlying theory? In this paper we present a systematic way to constrain a
wide range of non-Gaussian shapes, including general single and multi-field
models and models with excited initial states. We present a separable,
divergent basis able to recreate many shapes in the literature to high accuracy
with between three and seven basis functions. The basis allows shapes to be
grouped into broad "template classes", satisfying theoretically-relevant priors
on their divergence properties in the squeezed limit. We forecast how well a
Planck-like CMB survey could not only detect a general non-Gaussian signal but
discern more about its shape, using existing templates and new ones we propose.
This approach offers an opportunity to tie together minimal theoretical priors
with observational constraints on the shape in general, and in the squeezed
limit, to gain a deeper insight into what drove inflation.
