Not always, I would say, as in the case of a crate sitting on the bed of an accelerating truck, the crate being stationary with respect to the truck, but accelerating with respect to the ground. Static friction provides the force necessary to accelerate the crate at the same rate as the truck. There is no other force acting on the crate in that direction, so there is no force for the friction force to oppose. The direction of the friction force is opposite the direction it would move with respect to the truck bed if no friction was present.
Friction between tyres of a cycle and ground allow the forward movement of the cycle.