We　present　different　regularizations　and　numerical　methods　for　the　nonlinear　Schrodinger　equation　with　singular　nonlinearity　(sNLSE)　including　the　regularized　Lie-Trotter　time-splitting　(LTTS)　methods　and　regularized　Lawson-type　exponential　integrator　(LTEI)　methods.　Due　to　the　blowup　of　the　singular　nonlinearity,　i.e.,　f　(p)　=　pa　with　a　fixed　exponent　a　＜　0　goes　to　infinity　when　p　-＞　0+　(p　=　|lk|2　represents　the　density　with　lk　being　the　complex-valued　wave　function　or　order　parameter),　there　are　significant　difficulties　in　designing　accurate　and　efficient　numerical　schemes　to　solve　the　sNLSE.　In　order　to　suppress　the　round-off　error　and　avoid　blowup　near　p　=　0+,　two　types　of　regularizations　for　the　sNLSE　are　proposed　with　a　small　regularization　parameter　0　＜　e　MUCH　LESS-THAN　1.　One　is　based　on　the　local　energy　regularization　(LER)　for　the　sNLSE　via　regularizing　the　energy　density　F(p)　=　pa+1/(a　+　1)　locally　near　p　=　0+　with　a　polynomial　approximation　and　then　obtaining　a　local　energy　regularized　nonlinear　Schrodinger　equation　via　energy　variation.　The　other　one　is　the　global　nonlinearity　regularization　which　directly　regularizes　the　singular　nonlinearity　f　(p)　=　pa　to　avoid　blowup　near　p　=　0+.　For　the　regularized　models,　we　apply　the　first-order　Lie-Trotter　time-splitting　method　and　Lawson-type　exponential　integrator　method　for　temporal　discretization　and　combine　with　the　Fourier　pseudospectral　method　in　space　to　numerically　solve　them.　Numerical　examples　are　provided　to　show　the　convergence　of　the　regularized　models　to　the　sNLSE　and　they　suggest　that　the　local　energy　regularization　performs　better　than　directly　regularizing　the　singular　nonlinearity　globally.