  We study ideals of Borel type, including $k$-Borel ideals and $t$-spread
Veronese ideals. We determine their free resolutions and their homological
shift ideals. The multiplicity and the analytic spread of equigenerated
squarefree principal Borel ideals are computed. For the multiplicity, the
result is given under an additional assumption which is always satisfied for
squarefree principal Borel ideals. These results are used to analyze the
behaviour of height, multiplicity and analytic spread of the homological shift
ideals $HS_j(I)$ as functions of $j$, when $I$ is an equigenerated squarefree
Borel ideal.
