Brush, graze, glance, shave, skim are comparable when they mean to touch lightly in passing.
Brush implies a movement like the flick of a brush upon a surface: sometimes it suggests no more than an almost impalpable touching, but sometimes it suggests a light touching or rubbing that disperses something that it touches.
Graze implies the swift passage of a bullet or any rapidly moving object so that it touches a person or thing abrading the surface or, in the case of a person, the skin.
Glance (see also FLASH) basically implies a blow (as from a sword, a spear, or an ax) that owing to the hardness or resistance of what is struck turns aside or slips and so fails of its full effect; hence glance in its participial form glancing is often used to describe such a blow either in its course or effect.
Shave implies a touching as lightly and closely as a razor that passes over the face; although it comes near to graze, it carries no implication of abrasion but rather in some contexts suggests a dangerous approach or a narrow escape.
Skim (see also FLY) also implies a light touch in passing, but it never suggests the action of anything (as a bullet, a razor, or a weapon) that is even slightly dangerous; rather it suggests an avoidance of depths by someone or something that touches upon the surface or dips only into shallows (skim a book in reading).