  The late-time cosmological dynamics of disformal gravity are investigated
using dynamical systems methods. It is shown that in the general case there are
no stable attractors that screen fifth-forces locally and simultaneously
describe a dark energy dominated universe. Viable scenarios have late-time
properties that are independent of the disformal parameters and are identical
to the equivalent conformal quintessence model. Our analysis reveals that
configurations where the Jordan frame metric becomes singular are only reached
in the infinite future, thus explaining the natural pathology resistance
observed numerically by several previous works. The viability of models where
this can happen is discussed in terms of both the cosmological dynamics and
local phenomena. We identify a special parameter tuning such that there is a
new fixed point that can match the presently observed dark energy density and
equation of state. This model is unviable when the scalar couples to the
visible sector but may provide a good candidate model for theories where only
dark matter is disformally coupled.
