One can view a 2-parameter Brownian sheet { W( s, t); s, t⩾0} as a stream of interacting Brownian motions { W( s,•); s⩾0}. Given this viewpoint, we aim to continue the analysis of [J.B. Walsh, The local time of the Brownian sheet, Astérisque 52–53 (1978) 47–61] on the local times of the stream W( s,•) near time s=0. Our main result is a kind of maximal inequality that, in particular, verifies the following conjecture of [D. Khoshnevisan, The distribution of bubbles of Brownian sheet, Ann. Probab. 23 (2) (1995) 786–805]: As s→ 0 + , the local times of W( s,•) explode almost surely. Two other applications of this maximal inequality are presented, one to a capacity estimate in classical Wiener space, and one to a uniform ratio ergodic theorem in Wiener space. The latter readily implies a quasi-sure ergodic theorem. We also present a sharp Hölder condition for the local times of the mentioned Brownian streams that refines earlier results of [M.T. Lacey, Limit laws for local times of the Brownian sheet, Probab. Theory Related Fields 86 (1) (1990) 63–85; P. Révész, On the increments of the local time of a Wiener sheet, J. Multivariate Anal. 16 (3) (1985) 277–289; J.B. Walsh, The local time of the Brownian sheet, Astérisque 52–53 (1978) 47–61].