  Low background searches for astrophysical neutrino sources anywhere in the
sky can be performed using cascade events induced by neutrinos of all flavors
interacting in IceCube with energies as low as ~1 TeV. Previously, we showed
that even with just two years of data, the resulting sensitivity to sources in
the southern sky is competitive with IceCube and ANTARES analyses using muon
tracks induced by charge current muon neutrino interactions - especially if the
neutrino emission follows a soft energy spectrum or originates from an extended
angular region. Here, we extend that work by adding five more years of data,
significantly improving the cascade angular resolution, and including tests for
point-like or diffuse Galactic emission to which this dataset is particularly
well-suited. For many of the signal candidates considered, this analysis is the
most sensitive of any experiment. No significant clustering was observed, and
thus many of the resulting constraints are the most stringent to date. In this
paper we will describe the improvements introduced in this analysis and discuss
our results in the context of other recent work in neutrino astronomy.
