In the present work electrically conductive, flexible, lightweight carbon sponge materials derived from open-pore structure melamine foams are studied and explored. Hydrophobic and hydrophilic surface properties - depending on the chosen treatment conditions - allow the separation and storage of liquid chemical compounds. Activation of the carbonaceous structures substantially increases the specific surface area from similar to 4 m(2)g(-1) to similar to 345 m(2)g(-1), while retaining the original three-dimensional, open-pore structure suitable for hosting, for example, Ni catalyst nanoparticles. In turn the structure is rendered suitable for hydrogenating acetone to 2-propanol and methyl isobutyl ketone as well for growing hierarchical carbon nanotube structures used as electric double-layer capacitor electrodes with specific capacitance of similar to 40 F/g. Mechanical stress-strain analysis indicates the materials are super-compressible (>70% volume reduction) and viscoelastic with excellent damping behavior (loss of 0.69 +/- 0.07), while piezoresistive measurements show very high gauge factors (from similar to 20 to 50) over a large range of deformations. The cost-effective, robust and scalable synthesis - in conjunction with their fascinating multifunctional utility - makes the demonstrated carbon foams remarkable competitors with other three-dimensional carbon materials typically based on pyrolyzed biopolymers or on covalently bonded graphene and carbon nanotube frameworks.