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Are headings important?Yes. The reason they are important is simple. They provide a map of the content on the page both visually and in terms of usability.
By default in html, the heading tags are bold and provide a line break. A page full of dense copy with headings is a lot easier for a user to scan in order to understand the point of the page.
They also improve usability by enabling visitors to locate the section of the page which features relevant information to their query. This is great for disabled users who may want an overview of the page and also a great way of signposting your content to the search engines.
How should you deploy them?Each page should have an H1 heading. If you don't have one, you will not be marked down, but it's such a simple way to show the topic of the page to users and search engines that it would be silly not to include one. Multiple H1s are not going to harm your page, but they serve to confuse the topic of the page. If you are changing your topic so much that you require a second H1, consider starting a new page.
H2 headings should be used for each subsection of the page, and will obviously be a subsection of information on the main topic of the page as described in the H1. You can use as many H2s as you need to and if you want to add further subsections underneath your H2 then simply add an H3. You can nest as many headings as you like up to H6, but you should always make sure that you do nest them and don't skip numbers. Again, this is not critical, but it is good practise.
What tools can I use to check them?Overlooking the fact that they inexplicably use the American spelling of the "analyse," this tool from UK company The Escape is great for seeing an outline of your page:
This tool puts your heading system into perspective showing you exactly what level you've decided your headings should be with helpful colour coding and points out if you've missed out a level. Try plugging in the addresses of a few of your favourite sites and see how they match up.
I reckon I might offer my services out to these guys: