In this paper it is proved that every $3$-connected planar graph contains a path on $3$ vertices each of which is of degree at most $15$ and a path on $4$ vertices each of which has degree at most $23$. Analogous results are stated for $3$-connected planar graphs of minimum degree $4$ and $5$. Moreover, for every pair of integers $n\ge 3$, $ k\ge 4$ there is a $2$-connected planar graph such that every path on $n$ vertices in it has a vertex of degree $k$.