Singularity has several different meanings:

• mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved
• gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum
• technological singularity - a predicted point in the development of a civilization at which technological progress accelerates beyond the ability of present-day humans to fully comprehend or predict