Does heat transfer by convection happen mostly in solids, liquids, or gases?
Convection only happens with liquids and gases.
Here is a definition of convection :
"the movement caused within a fluid by the tendency of hotter and therefore less dense material to rise, and colder, denser material to sink under the influence of gravity, which consequently results in transfer of heat." (https://en.oxforddictionaries.com/definition/convection)
Convection only happens with fluids, materials that can flow. Liquids can flow (think of water) and gases can flow (think of wind). Solids are stuck in place so they can't flow and since they can't flow, there is no convection.
The way heat transfers through solids is conduction.
"In short, it is the transfer of heat through physical contact." (https://www.universetoday.com/82331/what-is-conduction/)
Conduction happens in solids but also in liquids and gases.