  We consider resonances for fourth order differential operators on the
half-line with compactly supported coefficients. We determine asymptotics of a
counting function of resonances in complex discs at large radius, describe the
forbidden domain for resonances and obtain trace formulas in terms of
resonances. We apply these results to the Euler-Bernoulli operator on the
half-line. The coefficients of this operator are positive and constants outside
a finite interval. We show that this operator does not have any eigenvalues and
resonances iff its coefficients are constants on the whole half-line.
