Recent likelihood asymptotics has produced highly accurate p-values for many very general contexts. The terminal formulas for producing these p-values can however have serious singularities in the neighbourhood of the maximum likeli- hood value. The singularities near the maximum likelihood value are downstream versions of those addressed by Daniels (1987) for the scalar saddlepoint context; he provided an approximate value at the singularity which involved a standard- ized third order cumulant. For a general statistical context we develop a third order bridge for the p-value formula at the maximum likelihood singularity in the case with no nuisance parameters, and a second order bridge at the singularity for the case with nuisance parameters. We also develop a third order graphical procedure for bridging the singularities; it handles both the cases without and with nuisance parameters.