Ultrafast state-selective dynamics of diatomic molecules in the electronic ground state under the control of infrared picosecond and femtosecond shaped laser pulses is investigated for the discrete vibrational bound states and for the dissociative continuum states. Quantum dynamics in a classical laser field is simulated for a one-dimensional nonrotating dissociative Morse oscillator, representing the local OH bond in the H₂O and HOD molecules. Computer simulations are based on two approaches - exact treatment by the time-dependent Schrödinger equation and approximate treatment by integro-differential equations for the probability amplitudes of the bound states only. Combination of these two approaches is useful to reveal mechanisms underlying selective excitation of the continuum states and above-threshold dissociation in a single electronic state and for designing optimal laser fields to control selective preparation of the high-lying bound states and the continuum states. Optimal laser fields can be designed to yield almost 100% seletive preparation of any prescribed bound state, including those close to the dissociation threshold. State-selective preparation of the highest bound state may be accompanied by the appearance of a quasi-bound molecular state in the continuum with the kinetic energy of the fragments being close to zero. The respective above-threshold dissociation spectrum contains an additional, zero-order peak. The laser-induced dissociation from selectively prepared high-lying bound states is shown to be very efficient, with the dissociation probability approaching the maximal value. Flexible tools of state-selective laser control are developed which enable one to achieve selective control of the dissociation spectra resulting in time-selective and space-selective control of the dissociation fragments.