Spontaneous discrete symmetry breaking always implies domain walls I've read several times that if a discrete symmetry is spontaneously broken, then there exist domain walls that interpolate between the different vacua. However, Weinberg says that if the former happens, then we may have these domain walls.
Is there any case such that we have spontaneous breaking of a discrete symmetry but we cannot have domain walls? How would this be proven?
 A: If we start in the unbroken phase, and if there are multiple degenerate vacua after spontaneously breaking the symmetry, generically we should have domain walls. The reason is that as we pass the phase transition, there will be random fluctuations in the field(s) that will make different values of the order parameter locally more favorable, so that different spatial regions settle in to different values of the order parameter below the critical temperature.
I can't say it any better than Kibble [1], so I will just quote:

For $T$ near $T_c$, there will be large fluctuations in $\phi$. Once $T$ has fallen well below $T_c$, we may expect $\phi$ to have settled down with a non-zero expectation value corresponding to some point on [the space of degenerate vacua]. No point is preferred over any other. As in an isotropic ferromagnet cooled below its Curie point the choice will be determined by whatever small fields happen to be present, arising from random fluctuations. Moreover this choice will be made independently in different regions of space, provided they are far enough apart. (What is far enough we shall discuss shortly.) Thus we can anticipate the formation of an initial domain structure with the expectation value of $\phi$, the order parameter, varying from region to region in a more or less random way.

Kibble also points out that we can circumvent this formation of domain structure if the universe not neutral, so that one vacuum is favored over the others. It's not that domain walls are logically necessary, just expected.
[1]: T W B Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9 1387 (1976)
