  We investigate the linearizability problem for different classes of 4-webs in
the plane. In particular, we apply a recently found in [AGL] the
linearizability conditions for 4-webs in the plane to confirm that a 4-web MW
(Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant
basic invariant is always linearizable (this result was first obtained in [M
28]); it also follows from the papers [Na 96] and [Na98]). Using the same
conditions, we also prove that such a 4-web with a constant basic invariant
(Nakai's web) is linearizable if and only if it is parallelizable. We also
study four classes of the so-called almost parallelizable 4-webs APW_a, a = 1,
2, 3, 4 (for them the curvature K = 0 and the basic invariant is constant on
the leaves of the web foliation X_a), and prove that a 4-web APW_a is
linearizable if and only if it coincides with a 4-web MW_a of the corresponding
special class of 4-webs MW. The existence theorems are proved for all the
classes of 4-webs considered in the paper.
