The scarcity or lack of access to essential services at the local and neighbourhood levels in cities can result in significant spatial inequalities, as some areas and their residents can deal with disadvantages and a lower quality of daily life. In particular, the spatial distribution and the variety of amenities at the local scale represent an important feature of the liveliness of places. The local availability and access to essential services are particularly relevant for some demographic groups experiencing limited mobility or mobility poverty, such as older adults living in cities, and spatial disparities have been further exacerbated by the COVID-19 pandemic, which highlighted severe difficulties in accessing essential services. This work explores the issue focussing on the following question: who can access what depending on where they live in cities? Using Machine Learning and Spatial Autocorrelation applied to different data sources for spatial information on the location of urban amenities and Internet access, this work aims to identify the most underserved places in terms of the variety of available amenities and access to quality broadband in three European capital cities. A comparison to urban areas where high percentages of older adults reside makes it possible to identify where residents can locally access several essential services (green spaces, health care, and local shopping) and where this need cannot be satisfied because of a lack in the amenity variety available at walking distance to their home. The combination of underserved areas with a high concentration of senior residents identifies left-behind areas in these cities, where interventions on inequalities are most needed. Results can inform policies aiming at favouring fair access to services at the local scale, possibly including slow and active mobility modes, and in general to develop comprehensive and sustainable planning strategies for cities, leaving no place and no person behind.