I used a friend's computer earlier to do something quick on this site and saw that none of my heading tags were appearing in his browser (Internet Explorer 9).
This was a bit distressing considering anyone browsing my site using IE 9 since it came on the block, so for about 4 months now, was seeing my site without any heading tags.
This means that none of my page or post titles were appearing anywhere on the site and any heading tags were invisible within the text of a post if I used them, as well.
Very confusing and unprofessional, indeed.
The good news is that the tags still technically are there, so the search engines can recognize this.
The reason you want the search engines to recognize this is because heading tags explicitly inform search engines of what content and more specifically what keywords are especially important and should be used to identify what your content is about.
You should be using heading tags as often as possible as a result.
On page SEO is still an important part of SEO and is simple to execute, so using your keywords within the heading tags should be part of your SEO strategy.
Remember to put your most targeted keywords in the H1 tag, then less important keywords in H2 and so on.
It's more of an aesthetic effect when it comes to using H1 tags for your readers as this typically just puts it in a more prominent font.
It's a problem if the heading tags aren't showing up AT ALL, however, as it just appears as blank space which can confuse your readers depending on how you're using the tags.
This is a problem which could be affecting a lot of WordPress based webmasters or anyone using Cufon for fonts.
Apparently it's a problem with Internet Explorer 9 not being compatible with some out of date versions of Cufon.
I was obviously one of those people, but fortunately the solution is simple enough to enact.
Download the latest cufon font js file from the Cufon website, then delete the old "cufon-yui.
js" file and add the new version to the wp-content/themes/yourtheme/js directory of your hosting account and this should fix the problem instantly.
This just goes to show the importance of routinely checking your website out in terms of appearance in different browsers using a free service like Browser Shots.