  We present an implementation of an optimal CMB trispectrum estimator which
accounts for anisotropic noise and incomplete sky coverage. We use a general
separable mode expansion which can and has been applied to constrain both
primordial and late-time models. We validate our methods on large angular
scales using known analytic results in the Sachs-Wolfe limit. We present the
first near-optimal trispectrum constraints from WMAP data on the cubic term of
local model inflation $ g_{\rm NL} = (1.6 \pm 7.0)\times 10^5$, for the
equilateral model $t_{\rm NL}^{\rm{equil}}=(-3.11\pm 7.5)\times 10^6 $ and for
the constant model $t_{\rm NL}^{\rm{const}}=(-1.33\pm 3.62)$. These results,
particularly the equilateral constraint, are relevant to a number of
well-motivated models (such as DBI and K-inflation) with closely correlated
trispectrum shapes. We also use the trispectrum signal predicted for cosmic
strings to provide a conservative upper limit on the string tension $G\mu \le
1.1\times 10^{-6}$ (at 95% confidence), which is largely background and model
independent. All these new trispectrum results are consistent with a Gaussian
Universe. We discuss the importance of constraining general classes of
trispectra using these methods and the prospects for higher precision with the
Planck satellite.
