I don’t have any specific suggestions other than they probably should be only slightly-larger than the normal font that is used on the site, and with less vertical margin space.   Variations in boldness, italic, serifs and so-on can be used to vary the tags without burning-up column inches.

h1

... on my computer, when used in-line as was done here, creates a 3/4" gap (yes, I measured it) in the text flow.   The character itself is 1/4" high, and there is a like-sized gap both above and below it.   The markup for what is shown here is this:   <h1>h1</h1> ... on my computer ...

On the one hand, that’s probably pretty-good markup for a document heading.   But, particularly on a small screen, that is a lot of real-estate being consumed gratuitously.