What the evidence says
The sources on this page do not measure household differences by rurality, heating fuel or vehicle ownership within income deciles directly. They give three partial views: transport's weight in household budgets, how housing-related spending changes across spending quintiles, and how a diesel tax increase passed through to prices in different regions.
On transport, Ivanova (2020) matched household survey data for 275,246 households to an emissions database. Land travel (vehicles, transport fuels and services) accounts for 32% of the carbon footprint of the top 10%. For the lowest expenditure quintile the paper reports a transport share of "about 20%", below only basic needs such as food and housing. The two figures do not use the same denominator, so they do not show whether transport weighs more heavily on lower-income households (see the next section). The design is cross-sectional and associational.
On housing, the same paper finds that the elasticity of housing-related spending decreases for the middle quintiles and increases for the highest quintile. The authors say this "may reflect" larger dwellings or shifts to single-family houses, and so higher heating and cooling needs among the highest spenders. They do not measure dwelling characteristics. The mechanism is an inference, and the evidence for it is medium at best.
The one causal design is Palanne (2018), a difference-in-differences study of a diesel tax increase in Finland. Pass-through rates were approximately 15 percentage points lower in the richest, wealthiest, most densely populated and most urban areas than at the other extremes. Pass-through was therefore higher in lower-income and more rural regions. The regional price gaps were mitigated over the following two to three years. The source does not quantify how much. The author concludes that, given the size of the tax raise, these regional differences likely tilted fuel taxation in a more regressive direction. That conclusion is an inference. The study did not measure distributional effects directly.
The page has no evidence on household heating fuel type, on rural versus urban vehicle dependence within an income decile, or on how large the cost differences are for households at the same income. These are the questions the topic asks, and the sources here do not answer them.