This paper investigates the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$ indexed by a permutation $w$. This number also equals the number of lattice points in the Newton polytope of $\mathfrak{S}_w(x)$. We establish a lower bound for this number in terms of the occurrences of patterns in $w$. The analysis is carried out in the general framework of dual characters of flagged Weyl modules. Our result considerably improves the bounds for principal specializations of Schubert polynomials or dual flagged Weyl characters previously obtained by Weigandt, Gao, and M{é}sz{á}ros--St. Dizier--Tanjaya. Some problems and conjectures are discussed.