  We study the occurrence of symmetry-enforced topological band crossings in
tetragonal crystals with strong spin-orbit coupling. By computing the momentum
dependence of the symmetry eigenvalues and the global band topology in the
entire Brillouin zone, we determine all symmetry-enforced band crossings in
tetragonal space groups. In particular, we classify all Dirac and Weyl
degeneracies on points, lines, and planes, and find a rich variety of
topological degeneracies. This includes, among others, double Weyl points,
fourfold-double Weyl points, fourfold-quadruple Weyl points, Weyl and Dirac
nodal lines, as well as topological nodal planes. For the space groups with
symmetry-enforced Weyl points, we determine the minimal number of Weyl points
for a given band pair and, remarkably, find that materials in space groups 119
and 120 can have band pairs with only two Weyl points in the entire Brillouin
zone. This simplifies the topological responses, which would be useful for
device applications. Using the classification of symmetry-enforced band
crossings, we perform an extensive database search for candidate materials with
tetragonal space groups. Notably, we find that Ba$_5$In$_4$Bi$_5$ and NaSn$_5$
exhibit twofold and fourfold Weyl nodal lines, respectively, which cross the
Fermi energy. Hf$_3$Sb and Cs$_2$Tl$_3$ have band pairs with few number of Weyl
points near the Fermi energy. Furthermore, we show that Ba$_3$Sn$_2$ has Weyl
points with an accordion dispersion and topological nodal planes, while AuBr
and Tl$_4$PbSe$_3$ possess Dirac points with hourglass dispersions. For each of
these candidate materials we present the ab-initio band structures and discuss
possible experimental signatures of the nontrivial band topology.
