Is it possible to conserve the total kinetic energy of a system, but not its momentum? It is possible to conserve momentum without conserving kinetic energy, as in inelastic
collisions. Is it possible to conserve the total kinetic energy of a system, but not its momentum? How?
To clarify, I am not necessarily talking about an isolated system. Is there any scenario which we could devise in which momentum is not conserved but kinetic energy is?
 A: Suppose that energy is conserved in one frame of reference, and you want it to be conserved in all other frames as well. Conservation of momentum is exactly the condition you need in order to make this happen in all frames.
To see this, consider what happens when you change to a different frame of reference, $v\rightarrow v+u$. Then all kinetic energies transform according to $K\rightarrow K+muv+\text{const.}$ (Potential energies don't change under this transformation.)
Let's say we write your question as a conjecture: --

If energy is conserved and total KE is conserved, then momentum is conserved.

Then your conjecture can actually be strengthened to read: --

If energy is conserved, then momentum is conserved.

(This is implicitly assuming that we want all frames of reference to be valid.)
A: In order for momentum to be conserved, it must be the case that $$\mathbf F_\text{net}=\frac{\text d\mathbf p}{\text dt}=0$$
In order for kinetic energy to be conserved, it must be the case that
$$\text dK=\text dW_\text{net}=\mathbf F_\text{net}\cdot\text d\mathbf x=0$$
at all instants in time.
So, is there a case where the net work done on an object is $0$, yet there is still a net force acting on the object? The answer is yes! We just need $\mathbf F_\text{net}\neq0$ to be perpendicular to the path of the object at all times. A simple example of this is an object undergoing uniform circular motion. The object's kinetic energy is not changing (as its speed remains constant), yet the momentum is constantly changing due to the non-zero net force.
A: 
Is there any scenario which we could devise in which momentum is not conserved but kinetic energy is?

When a ball bounces off the ground or a wall. The momentum is flipped but the kinetic energy stays about the same. 
A: If non-isolated system are of interest, then what you’re looking for is an external force that does no work. 


*

*The central force in a circular orbit: the satellites energy in unchanged, but its momentum is continuously changing. 

*An electron moving across a constant magnetic field: ditto
A: Yes, it is possible to conserve the total kinetic energy of the system but not the momentum.
Let me give you an intuitive explanation. 


*

*Suppose we have two charges, in which one is fixed and other is free to move and then they are released at some distance. In this case since one charge is fixed so the net external force on the system is not zero, so momentum of the system will not remain conserved, but kinetic energy + potential energy of the system will remain conserved, as there are no dissipative non conservative forces involved in the system. 

*Suppose a particle is tied to a point with the help of a string and is performing uniform circular motion on the horizontal surface. In this case the kinetic energy of the system will remain same but not the momentum.
