  We systematically explore the landscape of nonrelativistic effective field
theories with a local $S$-matrix and enhanced symmetries and soft behavior. The
exploration is carried out using both conventional quantum field theory methods
based on symmetry arguments, and recently developed on-shell recursion
relations. We show that, in contrary to relativistic theories, enhancement of
the soft limit of scattering amplitudes in nonrelativistic theories is
generally not a byproduct of symmetry alone, but requires additional low-energy
data. Sufficient conditions for enhanced scattering amplitudes can be derived
by combining symmetries and dispersion relations of the scattered particles.
This has direct consequences for the infrared dynamics that different types of
nonrelativistic Nambu-Goldstone bosons can exhibit. We then use a bottom-up
soft bootstrap approach to narrow down the landscape of nonrelativistic
effective field theories that possess a consistent low-energy $S$-matrix. We
recover two exceptional theories of a complex Schr\"odinger-type scalar, namely
the $\mathbb{C} P^1$ nonlinear sigma model and the
Schr\"odinger-Dirac-Born-Infeld theory. Moreover, we use soft recursion to
prove a no-go theorem ruling out the existence of other exceptional
Schr\"odinger-type theories. We also prove that all exceptional theories of a
single real scalar with a linear dispersion relation are necessarily
Lorentz-invariant. Soft recursion allows us to obtain some further general
bounds on the landscape of nonrelativistic effective theories with enhanced
soft limits. Finally, we present a novel theory of a complex scalar with a
technically natural quartic dispersion relation. Altogether, our work
represents the first step of a program to extend the developments in the study
of scattering amplitudes to theories without Lorentz invariance.
