On to the question, the
Universe, at least according to current theory, has no centre. The important thing to realise is that a 'singularity' is not a physical thing, i.e. you can't say 'oh look at that singularity over there'. In fact a singularity is merely a situation where our theories cease to be meaningful. The Big Bang model says that the
Universe was hotter and denser in the past, but current theory doesn't actually let us wind the clock back to the very beginning. The equations themselves stop being meaningful at some point.
This means that when we say 'the
Universe is 14 Billion years old' what we really mean is that it is 14 Billions years from the oldest time our theories can make an accurate prediction. At some point as we wind the clock back and the
Universe becomes more and more dense and hot we reach a point where we don't know how stuff behaves at those densities and temperatures. Our theories stop
working, and we describe this as a 'singularity' but that is a description of an equation, not a physical 'thing'.
Hopefully that answers part of your question. In terms of their being a 'centre', the Big Bang theory says that the
Universe is roughly the same everywhere, so there are no special points. A centre would be such a special point. Big Bang theory is compatible with the
Universe being infinite. So at the earliest time we can describe it with current theories, the
Universe was probably infinite in extent and everywhere very hot and dense. From this point, everything in the
Universe began moving away from everything else. This expansion is uniform, in the sense that at all points it appears to an observer at that point that they are at the centre, since the rate of expansion in all directions is the same. There is no universal centre.
The other thing to realise is that the Big Bang was not something that occurred at some particular place, sending material outwards into previously empty space.