  In this paper classical solutions of the degenerate fifth Painlev\'e equation
are classified, which include hierarchies of algebraic solutions and solutions
expressible in terms of Bessel functions. Solutions of the degenerate fifth
Painlev\'e equation are known to expressible in terms of the third Painlev\'e
equation. Two applications of these classical solutions are discussed, deriving
exact solutions of the complex sine-Gordon equation and of the coefficients in
the three-term recurrence relation associated with generalised Charlier
polynomials.
