Have a read through Did the Big Bang happen at a point? and the answers to it.
The singularity at the Big Bang is the zero time limit of the equation (the FLRW metric) that describes the expansion of the universe. Most physicists believe that this is a mathematical artefact and does not describe what actually happened. It seems likely that some quantum gravity effect becomes important at the incredibly high energies and densities near the Big Bang, and this prevented the singularity from occurring.
Singularities are common in mathematics. For example the simple function $y = 1/x$ is singular at $x = 0$ because the expression $1/0$ has an undefined value. Since we use mathematics to describe physical systems it's not unusual to find our theories predict singularities under some circumstances. In general the response to this is to believe that the mathematics we've used is an approximation and the predicted singularities would disappear if we used a more accurate mathematical model.
So the answer to your question is that the singularity probably didn't exist.