In a convolution model, we observe random variables whose distribution is the convolution of some unknown density $f$ and some known or partially known noise density $g$. We construct goodness-of-fit testing procedures, which are adaptive with respect to the smoothness parameter of unknown density $f$, and also (in some cases) to some unknown parameter of the noise density $g$. For known polynomially smooth noise density $g$, our adaptive procedures behave differently according to whether the density under the null hypothesis is polynomially or exponentially smooth. A payment for adaptation is noted in both cases and for computing this we provide a non-uniform Berry-Esseen type theorem for degenerate $U$-statistics. In the first case we prove that the payment for adaptation is optimal (thus unavoidable). For exponentially smooth noise density $g$ with symmetric stable law, we study a wider framework: a semiparametric model, where the self-similarity index $s$ of $g$ is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities $f$. In this context, we are able to provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, estimation of the functional $\int f^2$ and goodness-of-fit testing on $f$. These estimators are adaptive with respect to both self-similarity index $s$ of the noise density $g$ and smoothness parameter of $f$ and attain the rates which are known optimal for known noise distribution and fixed known smoothness parameter of $f$. As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities.