It Depends on Your Model
How closely do you want to model reality? The truth is, most collisions are some mixture of inelastic and elastic. (That is, momentum is transferred, but not always "cleanly," some of that energy gets transferred into deforming the objects.) You can see this sort of thing if you watch slow-motion videos of things striking other things; momentum is lost because of deformation. Physicists tend to talk about clay and billiard balls (or ball bearings) because these tend to approximate perfectly elastic or inelastic collisions. Other things, like a wet sponge hitting a roast chicken, is a good example of how most collisions are both elastic and not.
Anyways, you'll need more information about the colliding objects to do this successfully. Here are some solutions:
A Simple Solution
You give every object which can be involved in a potential collision a "squishy" attribute. If one or both things have the "squishy" attribute, then the collision follows the inelastic collision model. Otherwise, use the elastic model.
You can get more complicated about your solutions, but I think the mentioned one will work if you want to keep thing simple.
You could also put in a deformation factor on objects that could collide. Basically, it says that there is a certain percentage of the momentum that gets treated as elastic and the other percentage is inelastic. Some materials are one or the other, but most should be somewhere in between.
This is more real, but also more complicated. It's actually not super complicated, and people would notice the sweet physics in your game.
Deformation Factor + Hardness
You could also give those objects a hardness and a deformation factor. The hardness would put a limit on how much energy needs to be put in before the object starts deforming (or becomes a slightly inelastic collision). You can get energy from momentum and mass, but you may just know the mass and speed anyways. This means that, at slow speeds, some things will act elastically, but at higher speeds, that same item will just go splat and be inelastic.
This would be the most realistic of the proposed models here, but the complications could go unnoticed by the average person. After all, it models how stuff actually works really well, so most people won't notice. Some engineers and scientists would notice, and they may show it off to other engineers and scientists. It opens up some marketing channels, letting you possibly show it to minutephysics or vsauce on youtube; they and their audience like this sort of thing.