There are numerous applications of quantum tunnelling. A few that pop in my mind right now are:
Particles tunnel out of the nucleus of which they are bounded by a potential. Classically this is forbidden as the nucleus is very strongly being held together by strong nuclear forces.
Scanning tunneling microscope:
A scanning tunneling microscope is used to image atoms on the surface of an object. The way this is done is by getting a conducting tip very close to the surface of the object. When the tip is sufficiently close to the surface, electrons sometimes tunnel through from the surface to the conducting tip creating a measurable current. When a base/background current is established, the tip's position is varied and the surface atoms are modelled through changes in the current created.
Has a particle ever been observed while tunneling?
The answer is unfortunately no. This is impossible as particles are quantum objects they do not have the well defined trajectories we are used to from Classical Mechanics. A particle has a probability of being in a specific place at a particular time, and this probabiliy is described by the square of its wavefunction, i.e $|\psi(x, t)|^2$. The same applies to quantum tunneling. A particle has a certain probability of being observed inside (or outside) the classically forbidden region, and any measurements we make will only either observe a particle there or they will not observe it there. You simply cannot follow a particle's trajectory because quite frankly such a thing does not exist in Quantum Mechanics.