  We show that Brill--Noether loci in Hilbert scheme of points on a smooth
connected surface $S$ are non-empty whenever their expected dimension is
positive, and that they are irreducible and have expected dimensions. More
precisely, we consider the loci of pairs $(I, s)$ where $I$ is an ideal that
locally at the point $s$ of $S$ needs a given number of generators.
  We give two proofs. The first uses Iarrobino's descriptionof the
Hilbert--Samuel stratification of local punctual Hilbert schemes, and the
second is based on induction via birational relationships between different
Brill--Noether loci given by nested Hilbert schemes.
