  We consider the embedding capacity functions $c_{H_b}(z)$ for symplectic
embeddings of ellipsoids of eccentricity $z$ into the family of nontrivial
rational Hirzebruch surfaces $H_b$ with symplectic form parametrized by $b\in
[0,1)$. This function was known to have an infinite staircase in the monotone
cases ($b= 0$ and $ b= 1/3$). It is also known that for each $b$ there is at
most one value of $z$ that can be the accumulation point of such a staircase.
In this manuscript, we identify three sequences of open, disjoint, blocked
$b$-intervals, consisting of $b$-parameters where the embedding capacity
function for $H_b$ does not contain an infinite staircase. There is one
sequence in each of the intervals $(0,1/5)$, $(1/5,1/3)$, and $(1/3,1)$. We
then establish six sequences of associated infinite staircases, one occurring
at each endpoint of the blocked $b$-intervals. The staircase numerics are
variants of those in the Fibonacci staircase for the projective plane (the case
$b=0$). We also show that there is no staircase at the point $b=1/5$, even
though this value is not blocked. The focus of this paper is to develop
techniques, both graphical and numeric, that allow identification of potential
staircases, and then to understand the obstructions well enough to prove that
the purported staircases really do have the required properties. A subsequent
paper will explore in more depth the set of $b$ that admit infinite staircases.
