Typically, something like this:
- Row 1 (bar above the header): Free Shipping, or offer
- Row 2: Burger menu icon (left), logo (middle), cart icon (right).
- Row 3: Search box
When at the top of page:
It could be something like this:
What I’d recommend is keeping at least row 1 sticky, and row 2 (and optionally row 3) semi-sticky. Read more about sticky headers.
Amazon’s header on mobile is great inspiration:
- Row 1: burger menu, logo, account, cart
- Row 2: search box
- Row 3: (scrollable) link bar
Note that it contains a link bar below the search box, which features some of their categories. Read more on link bars.