The most simple example is the linear sigma model. It consists of four real scalar fields,
\Phi=(\sigma,\vec{\phi}).
It's Lagrangian reads
L=\frac{1}{2} (\partial_{\mu} \Phi) \cdot (\partial}_{\mu} \Phi) + \frac{\mu^2}{2} \Phi \cdot \Phi - \frac{\lambda}{4} (\Phi \cdot \Phi)^2.
I'm using the west-coast metric (+---), and thus the plus sign in front of the quadratic term is just of the opposite sign to be a proper mass term. What's going on is most easily seen already in the classical limit: The potential has a maximum rather than a minimum at \Phi=0, and thus you can not have \Phi=0 as a stable equilibrium condition. Quantizing the model, you cannot do perturbation theory around \langle \Phi \rangle=0.
You have to do perturbation theory around a minimum. This is continuous-fold degenerated since the model is symmetric under O(4) rotations in \Phi space, but you can chose any of these minima and do perturbation theory around that one. The usual convention is to take \langle \sigma \rangle=\sigma_0 and \langle \vec{\phi} \rangle=0.
The solution for the minimum for \sigma_0=\text{const} is given by
\mu^2 \sigma_0-\lambda \sigma_0^3=0, \Rightarrow \sigma_0=\frac{\mu}{\lambda}.
Now, you plug the ansatz
\Phi=(\sigma_0+\tilde{\sigma},\vec{\phi})
into the Lagrangian. You'll find that you get a theory, which describes one particle with positive squared mass m^2=2 \mu^2 and three particles with 0 mass.
This is one of the most simple examples of the spontaneous breaking of a global gauge symmetry. The model described above is the most simple effective model to describe pions, which appear here as the massless states. The vacuum is not symmetric under the full group O(4) anymore, but only under the O(3) rotations of the three last components of \Phi. The masslessness of the pions in this socalled chiral limit is exact to all orders in perturbation theory and is an example for the famous Nambu-Goldstone theorem: There are as many massless "Nambu-Goldstone bosons" in a theory with spontaneous symmetry breaking as the dimension of the symmetry group of the vacuum.
For more details on the linear sigma model, see my notes on QFT
http://theorie.physik.uni-giessen.de/~hees/publ/lect.pdf (p. 187ff)
