We show that single Higgs production in $����$ collisions through laser backscattering provides the best way to look for New Physics (NP) effects inducing anomalous Higgs couplings. Our analysis is based on the four $dim=6$ operators $��_{UB}$, $��_{UW}$, $\ol��_{UB}$ and $\ol��_{UW}$ which describe New Physics effects in this sector. Using the Higgs production rate we establish observability limits for the couplings of the aforementioned operators at a level of $10^{-3}$, which means establishing lower bounds on the scale of NP of the order 30 to 200 TeV. Higgs branching ratios, especially the ratio $��(H\to����)/��(H\to��Z)$, are shown to provide powerful ways to disentangle the effects of the various operators.

8 pages + 13 figures in ps file.