Does "natural" superposition of particles exist?

One of the common misconceptions that people starting out with QM often have is to think that a system is either in a superposition state or it is not. Actually superposition is only defined relative to a particular basis (such as the eigenstates of some observable).

If a system is in a state of superposition relative to one basis it is always possible to define a basis where it is not and vice versa.

So for example a particle with a definite position is in a superposition of momentum states or a spin pointing up relative to the $z$ direction is in a superposition of up and down relative to the $x$ or $y$ directions.

Yes, superposed states are ubiquitous. For example, the electrons in a molecule are not localized; this is a form of superposition. See Chemical Bonding as a Superposition Phenomenon for details, and also this blog post for a presentation of these ideas.

The canonical example is probably neutrino oscillations, where the flavor eigenstates are a superposition of the mass eigenstates, resulting in neutrinos changing flavors during the interaction-free evolution. The experimental verification and consequences of this were recognized by 1995 Nobel Prize in Physics.