  We consider noisy observations of a distribution with unknown support. In the
deconvolution model, it has been proved recently [19] that, under very mild
assumptions, it is possible to solve the deconvolution problem without knowing
the noise distribution and with no sample of the noise. We first give general
settings where the theory applies and provide classes of supports that can be
recovered in this context. We then exhibit classes of distributions over which
we prove adaptive minimax rates (up to a log log factor) for the estimation of
the support in Hausdorff distance. Moreover, for the class of distributions
with compact support, we provide estimators of the unknown (in general
singular) distribution and prove maximum rates in Wasserstein distance. We also
prove an almost matching lower bound on the associated minimax risk.
