  The Hausdorf moment problem (HMP) over the unit interval in an $L^2$-setting
is a classical example of an ill-posed inverse problem. Since various
applications can be rewritten in terms of the HMP, it has gathered significant
attention in the literature. From the point of view of regularization it is of
special interest because of the occurrence of a non-compact forward operator
with non-closed range. Consequently, HMP constitutes one of few examples of a
linear ill-posed problem of type~I in the sense of Nashed. In this paper we
highlight this property and its consequences, for example, the existence of a
infinite-dimensional subspace of stability. On the other hand, we show
conditional stability estimates for the HMP in Sobolev spaces that indicate
severe ill-posedness for the full recovery of a function from its moments,
because H\"{o}lder-type stability can be excluded. However, the associated
recovery %of the linear functional that characterizes of the function value at
the rightmost point of the unit interval is stable of H\"{o}lder-type in an
$H^1$-setting. We moreover discuss stability estimates for the truncated HMP,
where the forward operator becomes compact. Some numerical case studies
illustrate the theoretical results and complete the paper.
