Whenever we have "goosebumps", there is a momentary piloerection, due to which hair on our elbows and legs get "erected" and "stand" almost perpendicular to the surface of our skin. But why do our "head hair" not get erected ?
Piloerection of head hair does happen as often as it occurs elsewhere on the body, but head hair length tends to make it difficult to see. Close observation, especially with short hair, should make it visible.
Piloerection of head hair is something more commonly felt by the person it is happening to rather than seen by others (or in mirrors); when you get that "goose-bump" sensation across your scalp, that will be due to pilo-erection.
(PS added) I haven't provided citations - this is not a phenomena that has had much specific academic research and I have not found any stating it occurs - or stating it does not occur (except in association with balness) despite looking, but is almost universally experienced, often without close examination or reflection about what specifically happens.
Citing references for baldness resulting in absence of piloerection on the scalp is not quite the same as a reference specifically stating it happens, but ... Professor Rodney Sinclair, head of dermatology at the University of Melbourne says - “You can’t get goose bumps on the scalp when you go bald and you can’t regrow hairs either because the follicles can’t regenerate.” ( https://www.futurity.org/goose-bump-muscle-1324242-2/ ) ( https://link.springer.com/article/10.1007%2Fs12565-016-0359-5 )
Many sources say that it does. See the word "horripilation", for instance. Or no less an authority than Shakespeare, who tells us
I could a tale unfold whose lightest word Would harrow up thy soul, freeze thy young blood, Make thy two eyes, like stars, start from their spheres, Thy knotted and combined locks to part And each particular hair to stand on end, Like quills upon the fretful porpentine
(For those unfamiliar with English Literature, the source is William Shakespeare's play "The Tragedy of Hamlet, Prince of Denmark", Act I, scene 5, lines 15-20, first performed ca 1602, and published in various versions between 1603 and 1623: https://en.wikipedia.org/wiki/Hamlet )