We study experimentally the thermal fluctuations of energy input and dissipation in a harmonic oscillator driven out of equilibrium, and search for fluctuation relations. Both the transient evolution from the equilibrium state, and non-equilibrium steady states are analyzed. Fluctuation relations are obtained experimentally for both the work and the heat, for the stationary and transient evolutions. A stationary state fluctuation theorem is verified for various time dependences of the imposed external torque. The transient fluctuation theorem is satisfied for the work given to the system but not for the heat dissipated by the system in the case of linear forcing. Experimental observations on the statistical and dynamical properties of the position fluctuations of the torsion pendulum allow us to derive analytical expressions for the probability density functions of the work and the heat. We obtain for the first time an analytic expression for the probability density function of the heat. The agreement between experiments and our predictions is excellent.