  This is a survey paper discussing one specific (and classical) system of
algebraic equations - the so called "Prony system". We provide a short overview
of its unusually wide connections with many different fields of Mathematics,
stressing the role of Singularity Theory. We reformulate Prony System as the
problem of reconstruction of "Spike-train" signals of the form
$F(x)=\sum_{j=1}^d a_j\delta(x-x_j)$ from the noisy moment measurements. We
provide an overview of some recent results of [1-3, 6, 8, 9, 11, 12, 5] on the
"geometry of the error amplification" in the reconstruction process, in
situations where the nodes $x_j$ near-collide. Some algebraic-geometric
structures, underlying the error amplification, are described (Prony, Vieta,
and Hankel mappings, Prony varieties), as well as their connection with
Vandermonde mappings and varieties. Our main goal is to present some promising
fields of possible applications of Singulary Theory.
