  We demonstrate that there appear finite-time future singularities in $f(T)$
gravity with $T$ being the torsion scalar. We reconstruct a model of $f(T)$
gravity with realizing the finite-time future singularities. In addition, it is
explicitly shown that a power-low type correction term $T^\beta$ ($\beta>1$)
such as a $T^2$ term can remove the finite-time future singularities in $f(T)$
gravity. Moreover, we study $f(T)$ models with realizing inflation in the early
universe, the $\Lambda$CDM model, Little Rip cosmology and Pseudo-Rip
cosmology. It is demonstrated that the disintegration of bound structures for
Little Rip and Pseudo-Rip cosmologies occurs in the same way as in gravity with
corresponding dark energy fluid. We also discuss that the time-dependent matter
instability in the star collapse can occur in $f(T)$ gravity. Furthermore, we
explore thermodynamics in $f(T)$ gravity and illustrate that the second law of
thermodynamics can be satisfied around the finite-time future singularities for
the universe with the temperature inside the horizon being the same as that of
the apparent horizon.
