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 $sim0.3Re<{rm R}_{pl}<12Re$ 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.
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