Numerous combustion applications are concerned with the stabilization of diffusion flames formed by injecting gaseous fuels into a co-flowing stream containing an oxidizer. The smooth operation of these devices depends on the attachment and lift-off characteristics of the edge flame at the base of the diffusion flame. In this paper, we address fundamental issues pertinent to the structure and dynamics of edge flames, which have attributes of both premixed and diffusion flames. The adopted configuration is the mixing layer established in the wake of a splitter plate where two streams, one containing fuel and the other oxidizer, merge. The analysis employs a diffusive-thermal model which, although it excludes effects of gas expansion, systematically includes the influences of the overall flow rate, unequal strain rates in the incoming streams, stoichiometry, differential and preferential diffusion, heat loss and gas–solid thermal interaction, and their effect on the edge structure, speed, and temperature. Conditions when the edge flame is anchored to the plate, lifted-off and stabilized in the flow, or blown-off, are identified. Two stable modes of stabilization are observed for lifted flames; the edge flame either remains stationary at a specified location or undergoes spontaneous oscillations along a direction that coincides with the trailing diffusion flame.