Let TBP be the server busy period of an M/G/1 queueing system characterized by arrival intensity λ and service time c.d.f. A(τ). In this paper, we investigate the regularity structure of the Laplace transform σBP(s)=E[] on the complex s-plane. It is shown, under certain broad conditions, that finite singular points of σBP(s) are all branch points. Furthermore the branch point s0 having the greatest real part is always purely negative and is of multiplicity two. The basic branch point s0 and the associated complex structure provide a basis for an asymptotic representation of various descriptive distributions of interest. For a natural relaxation time |s0|-1 of the M/G/1 system, some useful bounds are obtained and the asymptotic behavior as traffic intensity approaches one is also discussed. Detailed results of engineering value are provided for two important classes of service time distributions, the completely monotone class and the Erlang class.