The discovery of two-dimensional (2D) iron monolayer in graphene pores stimulated experimental and computational material scientists to investigate low-dimensional elemental metals. There have been many advances in their synthesis, stability, and properties in the last few years. Inspired by these advancements, we investigated the electronic structure and elasticity of free-standing monolayers of group 10 elemental metals, viz. Ni, Pd, and Pt. Using density-functional theory (DFT), we explored the energetic, geometric, electronic, and elastic properties of hexagonal, honeycomb, and square lattice structures of each element, in both planar and buckled forms. Among planar configurations, the order of increasing stability is honeycomb, square, and hexagonal. In buckled form, this ordering remains the same for Pt but is reversed for Ni and Pd. Upon geometrical optimization, the extent of buckling for Pt was found to be small compared to Ni and Pd. The effect of buckling on the electronic structure was further scrutinized through the projected density of states, and it was found that highly buckled configurations derive their of states from 3D bulk, which highlights the correlation between buckled configurations and 3D bulk. For Pt in buckled square and honeycomb lattices, the density of states correlates more closely to their 2D monolayers. Regarding elasticity, the in-plane elastic constants indicate that all planar and buckled square lattices are unstable.