I use the same liquids I use to thin the paints. For acrylics, I will use water or isopropyl. For enamels and oils, I will use mineral spirits or lacquer thinner. I dip the brush in the liquid, tap it against the side of the jar, and then draw the brush across a paper towel. I repeat this, till the bristles are clean.
If the nub or root of the brush gets clogged, I've also user SuperClean, in place of the other liquids.
After cleaning the brush, I will shape the bristles to preserve the shape. Then I place them in my holders, upright, and let them air-dry. For holders, I have a wooden block with holes drilled in it, and an old ceramic toothbrush caddy.
I have not noticed any issue with the bristles and the potential harshness of some of these liquids. I have noticed, however, definite issues with the make of the brush. We all understand that good brushes are more expensive up front, but that it's a cost amortized over time, because quality brushes last a long time. But I've found that some brands made in the PRC, while using sable bristles, have tended to curl, right out of the package, rendering them useless. It pays to stick with the well-known brands.