It has recently become possible (Novoselev, 2005) to extract a single sp2-bonded carbon plane (commonly known as graphene) from a graphite substrate, and to measure its electrical properties. The characteristic zero electronic energy gap predicted by theory is consistent with measured electrical properties of these isolated graphene planes, and the quality of the samples is good enough that the quantum Hall effect can be observed (Zhang et al., 2005). Carbon nanostructures such as these are attracted to substrates and to other nanosystems by dispersion (van der Waals) forces. The present work further explores the very recent predictions of surprising power laws E = -KD-P for the energy versus separation curves of some of these systems when separated by a nonbonding distance D, with exponents P different from the conventional values. For example, for two perfect undoped graphene planes at zero temperature, in parallel orientation separated by distance D, the new analysis in (Dobson et al., 2006) revealed a vdW energy per unit area of form -CD-3. By contrast, previously accepted theory had predicted -KD-4, qualitatively understandable from a sum of R-6 contributions from individual atoms or bonds separated by a variety of distances if. The present work investigates the attraction of graphene to three-dimensional metallic substrates, covering both the undoped T = 0K graphene plane where -CD-3 applies, and cases of finite temperature and/or doping where a CD-5/2 law, characteristic of a two-dimensional electron gases (Dobson et al., 2005 and Sernelius and Bjork, 1998), are found. For low temperatures and light doping, both laws probably contribute. Direct measurement of these weak but surprisingly long-ranged forces should be aided by the capability developed by several groups (Ono et al., 2003) to measure forces on a sub-piconewton scale.