During oscillatory motion, whether it's a simple pendulum, a block on a spring, or liquid sloshing back and forth in a well...
Let's take a spring because it makes the argument clear.
It's important to understand that the spring and the mass on the end of it are not a closed system. That's because the top end of the spring is fixed to something, and we normally just take that as a fixed point. The fixed point exerts a force on the combined spring/mass system, and the force means that the momentum of the spring and mass is not conserved.
Is energy conserved? Well the top of the spring is fixed i.e. it doesn't move, so it can't do any work on the system, and that means that energy is conserved.
So momentum is not conserved though energy is. But this is not a general propery of an oscillator. It happens only because we're not dealing with a closed system. To illustrate this consider the following oscillator:
This is a closed system because we have just the spring and two masses, and there are no external connections. In this oscillator both energy and momentum are conserved (in fact the momentum is zero). Energy is conserved because there are no external forces acting so no work can be done on the system. Momentum is conserved because the forces acting on the two identical masses are equal and opposite, so the rate of change of momentum of the two masses is equal and opposite, and cancels out leaving zero.
The same arguments apply to the pendulum, though they are a little more obscure as the geometry is more complicated. However, the liquid sloshing to and fro is a bit different. If the liquid and the container are isolated, e.g. a ball floating in space with the liquid sloshing around inside it, then momentum is conserved because no external forces are acting. However the viscous losses in the liquid will convert the kinetic energy of the liquid to heat. So at a first glance it looks as if energy is not conserved. However if you count the increase of internal energy due to the temperature rise you'll find hat energy is still conserved.
Conservation of energy and momentum (and lots of other things) are intimately related to the symmetry of the system by Noether's theorem. Energy is always conserved if a system is unaffected by a displacement in time, and momentum is always conserved if the system is unaffected by a displacement in space. If you look at the first picture, then if you only include the spring and mass it's obvious that the system is not symmetric to displacement in space because the spring/mass would be in a different place wrt the fixed point, and that's why momentum isn't conserved. To conserve momentum you'd have to include whatever the fixed point is connected to.
I. Conservation of energy and momentum
If the system is isolated, energy will be conserved. If no external force acts on the system, momentum will be conserved. These are the laws of conservation of energy and conservation of momentum, respectively. You can find proofs for these laws within the context of Newtonian physics in any textbook on classical mechanics. Notice the importance of deciding what is the system of interest. For the same physical process, in one system momentum (for example) will be conserved and in another it won't.
Let's examine one example of oscillatory motion in detail.
Suppose we have a mass attached to an ideal spring attached to an infinitely massive wall oscillating horizontally on a frictionless plane. The system is the mass and the spring. This system is isolated---the wall is infinitely massive, so its motion isn't affected by spring force. The spring is ideal so it doesn't even warm up the wall. In fact, it is straightforward to solve for the motion of the mass and then to calculate the energy explicitly and see that it is a constant. The momentum of the system will clearly not be conserved since the velocity of the mass is a function of time. In fact, there is an external force acting on the system, so we shouldn't expect it to be conserved.
Now assume the spring to be connected to some (possible very large) mass that also rests on the frictionless plane. Let's consider the system of interest to include both masses and the spring. If we assume this new system is isolated, energy will again be conserved. As before, one can calculate the energy explicitly to see that this is true. If we consider the total momentum of both masses taken together, momentum will now be conserved as long as no external force acts on the system.
II. Motion near the equilibrium position
Motion is described vectorially, so it is natural to think in terms of momentum here. For small displacements near equilibrium momentum will be constant since the spring force at equilibrium is vanishingly small. This is just Newton's first law. Thus, the mass just passes by the equilibrium position with its momentum essentially unaffected.
If you prefer to think in terms of energy, the kinetic energy of the mass as it passes through the equilibrium position will also be constant since the change in the potential is vanishingly small. Thus, the object's speed must remain essentially constant as it passes through equilibrium.
The energy is conserved if the external force does not depend on time explicitly. It follows from the Newton equation multiplied by velocity (increment of the kinetic energy is compensated by the decrement of the potential energy).
The momentum is conserved when the force equals zero. Again, the Newton equation says the momentum increment is due to the external force.
It is the force who is a driving factor.
