Singularities in turbulence
What is the concept of singularity in turbulence? Please explain it.

As long as i understand your question, singularity in turbulence is related to the mathematical model we use for fluid flows (i.e., NavierStokes equations) and numerical simulations based upon it.
The fact is that for Re numbers above some (quite low) critical limit, the solution to NS equations can't be proven to be bounded in some norm (AND to be unique) and it actually blows up, you have a singularity in a finite amount of time (your simulation ends up with NaN). As i understand this matter (please correct me if i'm wrong), this is related to turbulence by the fact that singularities have to emerge at the lowest scales, where energy is cascaded by turbulence. What the real connection is with turbulence, well i don't actually know it...but the fact that several (non unique) weak solutions exists in this case, i think it's quite suggestive. Of course, no actual singularity exists in reality 
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