  A linear layout of a graph $ G $ consists of a linear order $\prec$ of the
vertices and a partition of the edges. A part is called a queue (stack) if no
two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v
\prec x \prec y \prec w $ ($ v \prec x \prec w \prec y $) may not be in the
same queue (stack). The best known lower and upper bounds for the number of
queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42
[Bekos et al., Algorithmica 2022], respectively. While queue layouts of special
classes of planar graphs have received increased attention following the
breakthrough result of [Dujmovi\'c et al., J. ACM 2020], the meaningful class
of bipartite planar graphs has remained elusive so far, explicitly asked for by
Bekos et al. In this paper we investigate bipartite planar graphs and give an
improved upper bound of 28 by refining existing techniques. In contrast, we
show that two queues or one queue together with one stack do not suffice; the
latter answers an open question by Pupyrev [GD 2018]. We further investigate
subclasses of bipartite planar graphs and give improved upper bounds; in
particular we construct 5-queue layouts for 2-degenerate quadrangulations.
