How cities can become more conducive to sustainable travel patterns is a pressing matter in urban research. In the built environment–travel literature, several features potentially promoting sustainable mobility remain under-researched. One concerns the precise role of local accessibility – in terms of the number, variety and specific types of amenities – in promoting the slow modes of walking and cycling and in reducing car use. Another is the possibility that the standard assumption of a linear relationship between density and (un)sustainable travel is flawed. To fill these gaps, we explore specific local amenity supply types that stimulate a large share of residents to walk or cycle, avoiding long drives, to accomplish their daily activities. In a case study of the Västra Götaland region in Sweden, we analyse a combination of high-quality micro data from registers and the national travel survey using a stepwise statistical approach. We find evidence of non-linear discontinuities and significant geographical variation in how local accessibility affects travel behaviour. While more neighbourhood amenities contribute to more walking/cycling, the effect of establishing additional amenities varies amongst neighbourhood types and travel variables. Local everyday services, for example, grocery stores and preschools, are critical in triggering change. The findings inform urban planning by showing how a basic supply of essential amenities could contribute to better prerequisites for walking or cycling and reduced car dependence by inducing residents to choose local options.