While stochastic parabolicity condition is well-known, there is no immediate analogue for stochastic equations that are second-order in time. The objective of this paper is to present a general approach to the study of well-posedness and stability in Sobolev spaces on $bRd$ of the initial value problem for second-order in time stochastic evolution equations with constant coefficients and multiplicative time-only Gaussian white noise, and to illustrate the results on some particular examples. In general, there is no single condition for well-posedness in terms of the coefficients of the equation, but analysis of certain classes of equations leads to analogues of the stochastic parabolicity condition.