  The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous
non-differentiable function constructed by Takagi in 1903. This paper studies
the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It
shows that for a full Lebesgue measure set of ordinates y, these level sets are
finite sets, but whose expected number of points is infinite. Complementing
this, it shows that the set of ordinates y whose level set has positive
Hausdorff dimension is itself a set of full Hausdorff dimension 1 (but Lebesgue
measure zero). Finally it shows that the level sets have a nontrivial Hausdorff
dimension spectrum. The results are obtained using a notion of "local level
set" introduced in a previous paper, along with a singular measure
parameterizing such sets.
