Is there any way to tell if the Big Bang happened everywhere or just in some regions of the entire universe? On large scales, matter seems to spread uniformly in our observable universe so we think the Big Bang happened everywhere in the observable universe.
Is there any way to tell if the Big Bang happened everywhere simultaneously in the entire universe or just in some regions of space?
 A: When it comes to the visible universe, the "concordance model" seems to fit observations very well. This concordance model is a standard Big Bang model with (very close to) zero curvature (flat geometry) and currently about 70% dark energy (acceleratory) and 30% matter (deceleratory), and with inflation (rapid acceleration) in the first fraction of a second.
In this model, the universe is uniform with only tiny variations in the cosmic background radiation, and a slow clumping of matter into galaxy clusters, etc. These variations are presumably of a quantum origin during early inflation, within "the same" Big Bang. Within this Big Bang, it all happened "at once" in the sense that Friedmann models (when you study General Relativity) can define a cosmic time, t that uniformly ticks forward.
However, we know little of the inflationary part, and there are many inflation theories. In some, there is a "phase transition" from the inflaton field at some event (time and location), and there is a local Big Bang bubble, resulting in our non-inflating universe, which may be much larger than the visible part. Outside this universe of ours, inflation continues, but similar phase transitions may occur at multiple points, resulting in many, causally disconnected universes, a "multi-verse." The Friedmann model would likely need some modification near the inflaton-Big Bang bubble wall -- perhaps space is not fully isotropic there, perhaps cosmic time is not easily defined -- but if our visible portion is far from the bubble wall, then a Friedmann model seems to be a very good approximation, according to observations.
The image below is a bit pop-sciency, but I like it, other than the spacing between universes would be huge.

