Header search
Appearance → Customize → Header → Display the search icon. On by default.
How it behaves
On desktop the bar carries a magnifying-glass button. Clicking it drops a panel below the icon with a normal WordPress search form in it; clicking again (or the close icon it swaps to) puts it away.
On a phone there is no room for a reveal, so the form lives inside the mobile menu instead, always open, at the top of the panel. The same switch governs both.
The search itself is plain WordPress: it submits to ?s=, and the results are
rendered by the theme’s search template with the
matched words highlighted in each excerpt.
Turning it off
Untick Display the search icon. The icon leaves the bar and the form leaves the mobile panel — one switch, both places, for the same reason the social links work that way: the mobile panel is where the bar’s contents go once the bar collapses, so leaving the form behind would read as a bug.
Search still works: /?s=your+term is a WordPress URL, and a
search widget in the sidebar still
functions. You have only removed it from the header.
JavaScript
The reveal is the one part of the theme that genuinely needs JavaScript, so with JS disabled the whole block is hidden rather than left as a button that does nothing. The mobile panel’s copy is a plain always-open form and needs no script at all.
The results page
Search results use their own compact result markup rather than the blog’s display modes, and always show an excerpt — even when Post text is set to full content, since a page of whole posts is the opposite of what someone scanning results wants.
The column layout is shared: search follows Blog → Layout, so a sidebar on your archives appears on your search results too.