  In this paper we treat the $\tau$-tilting finiteness of biserial
(respectively special biserial) algebras over algebraically closed
(respectively arbitrary) fields. Inside these families, to compare the notions
of representation-finiteness and $\tau$-tilting finiteness, we reduce the
problem to the $\tau$-tilting finiteness of minimal representation-infinite
(special) biserial algebras. Building upon the classification of minimal
representation-infinite algebras, we fully determine which minimal
representation-infinite (special) biserial algebras are $\tau$-tilting finite
and which ones are not. To do so, we use the brick-$\tau$-rigid correspondence
of Demonet, Iyama and Jasso, and the classification of minimal
representation-infinite special biserial algebras due to Ringel.
  Furthermore, we introduce the notion of minimal $\tau$-tilting infinite
algebras, analogous to the notion of minimal representation infinite algebras,
and prove that a minimal representation-infinite (special) biserial algebra is
minimal $\tau$-tilting infinite if and only if it is a gentle algebra. As a
consequence, we conclude that a gentle algebra is $\tau$-tilting infinite if
and only if it is representation infinite. We also show that for every minimal
representation-infinite (special) biserial algebra, the notions of tilting
finiteness and $\tau$-tilting finiteness are equivalent. This implies that a
mild (special) biserial algebra is tilting finite if and only if it is brick
finite.
