The two-dimensional boundary-layer flow over a cooled/heated flat plate is investigated.A cooled plate (with a free-stream flow and wall temperature distributionwhich admit similarity solutions) is shown to support non-modal disturbances, whichgrow algebraically with distance downstream from the leading edge of the plate. Ina number of flow regimes, these modes have diminishingly small wavelength, whichmay be studied in detail using asymptotic analysis.Corresponding non-self-similar solutions are also investigated. It is found that thereare important regimes in which if the temperature of the plate varies (in such a wayas to break self-similarity), then standard numerical schemes exhibit a breakdown ata finite distance downstream. This breakdown is analysed, and shown to be relatedto very short-scale disturbance modes, which manifest themselves in the spontaneousformation of an essential singularity at a finite downstream location. We show howthese difficulties can be overcome by treating the problem in a quasi-elliptic manner,in particular by prescribing suitable downstream (in addition to upstream) boundaryconditions.