Many works have considered two-dimensional free-surface flow over the edge of a horizontal plate, forming a waterfall, and with uniform horizontal flow far upstream. The flow is assumed to be steady and irrotational, whilst the fluid is assumed to be inviscid and incompressible. Gravity is also taken into account. In particular, amongst these works, numerical solutions for supercritical flows have been computed, utilising conformal mappings as well as a series truncation and collocation method. Here, an extension to this work is presented where a more appropriate expression is taken for the assumed form of the complex velocity. The justification of this lies in the behaviour of the flows far downstream and the wish to better encapsulate the parabolic nature of such a free-falling jet. New numerical results will be presented, demonstrating the improved shape of the new free-surface profiles. Further adjustments to the method are presented which lead to enhanced coefficient decay. The aforementioned adjustments are also applied to other supercritical flows (such as weir flows) and similar improvements to the jet shape can be observed. Flows that are still horizontal upstream but instead negotiate a convex corner and then run along an angled supporting bed (i.e. spillway flows) are also surveyed. New spillway problems and results are presented, where the spillway’s angled wall is more complex than a linear path; and, again, series truncation and collocation are utilised. Finally, a wake model for potential flow past a finite plate, perpendicular to the oncoming flow and below a free surface, is pursued. The approach here is to adopt a closure model of horizontal flow far downstream and use the boundary integral equation method to obtain a solution numerically. Related free-boundary problems are included to progress from a case of zero-gravity, unbounded flow to the full problem.