  We continue developing the theory of conical and vertical square functions on
$R^{n}$, where $\mu$ is a power bounded measure, possibly non-doubling. We
provide new boundedness criteria and construct various counterexamples.
  First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$
type testing conditions to characterise the $L^{2}$ boundedness. Second, we
completely answer the question, whether the boundedness of our operators on
$L^{2}$ implies boundedness on other $L^{p}$ spaces, including the endpoints.
For the conical square function, the answers are generally affirmative, but the
vertical square function can be unbounded on $L^{p}$ for $p > 2$, even if $\mu
= dx$. For this, we present a counterexample. Our kernels $s_t$, $t > 0$, do
not necessarily satisfy any continuity in the first variable -- a point of
technical importance throughout the paper.
  Third, we construct a non-doubling Cantor-type measure and an associated
conical square function operator, whose $L^{2}$ boundedness depends on the
exact aperture of the cone used in the definition. Thus, in the non-homogeneous
world, the 'change of aperture' technique -- widely used in classical tent
space literature -- is not available. Fourth, we establish the sharp
$A_{p}$-weighted bound for the conical square function under the assumption
that $\mu$ is doubling.
