The authors consider the scalar linear stochastic differential equation
where the time lag is a continuous function vanishing at infinity and where is a standard Brownian motion. Depending on the decay of to zero as , the solution process is proved to be almost surely oscillatory or nonoscillatory. The key ingredients for the proof are a random functional-differential equation solved by with a geometric Brownian motion as coefficient and related results for deterministic functional-differential equations. In addition, the long-time asymptotics of the solutions are studied in detail.