  We have performed extensive simulations to explore the possibility of
detecting eclipses and transits of close, sub-stellar and planetary companions
to white dwarfs in WASP light-curves. Our simulations cover companions
$\sim0.3\Re<{\rm R}_{pl}<12\Re$ and orbital periods $2{\rm h}<P<15{\rm d}$,
equivalent to orbital radii $0.003{\rm AU} < a < 0.1{\rm AU}$. For Gaussian
random noise WASP is sensitive to transits by companions as small as the Moon
orbiting a $\textrm{V}\simeq$12 white dwarf. For fainter white dwarfs WASP is
sensitive to increasingly larger radius bodies. However, in the presence of
correlated noise structure in the light-curves the sensitivity drops, although
Earth-sized companions remain detectable in principle even in low S/N data.
Mars-sized, and even Mercury-sized bodies yield reasonable detection rates in
high-quality light-curves with little residual noise. We searched for eclipses
and transit signals in long-term light-curves of a sample of 194 white dwarfs
resulting from a cross-correlation of the McCook $&$ Sion catalogue and the
WASP archive. No evidence for eclipsing or transiting sub-stellar and planetary
companions was found. We used this non-detection and results from our
simulations to place tentative upper limits to the frequency of such objects in
close orbits at white dwarfs. While only weak limits can be placed on the
likely frequency of Earth-sized or smaller companions, brown dwarfs and gas
giants (radius $\approx \Rjup$) with periods $<0.1-0.2$ days must certainly be
rare ($<10%$). More stringent constraints likely requires significantly larger
white dwarf samples, higher observing cadence and continuous coverage. The
short duration of eclipses and transits of white dwarfs compared to the cadence
of WASP observations appears to be one of the main factors limiting the
detection rate in a survey optimised for planetary transits of main sequence
stars.
