How to diagnose your site with google advanced search

Here are few suggestions:

Make sure your site is free from any type of filtering issues i.e. search [yourdomain.com] in Google; if your sites comes up #1 for this query, you are doing alright. If not, that’s a bad signal.

Make sure there are no duplicate content problems: search for a pretty long citation from your site. With a blog, for example, you have a good chance to see the category page instead of the corresponding post page.

Check how many URLs from your site have been indexed:
search [site:yourdomain.com] and “dig deeper” into the search results:
[site:yourdomain.com/subdirectory1/2/3 ..etc]

Learn if the site has canonical problems (for sites using www): search [site:yourdomain.com -inurl:www] and see if any non-www URLs have been stored in the index).

Identify most powerful pages of your site:
[ www site:yourdomain.com]
[ tld site:yourdomain.tld]
[inurl:domain site:yourdomain.com]
[domain site:yourdomain.com]

Identify most powerful pages of your site (keyword-dependent): search [site:yourdomain.com inanchor:keyword]

Find sites with most potential:
[site:yourdomain.com inanchor:"key * word"]
[site:yourdomain.com intitle:"key * phrase"]

Find most relevant pages of your site (to further promote them for the specified term): search [site:yourdomain.com keyword] or [site:yourdomain.com key * phrase]

Check your site is crawled and indexed frequently enough: search [site:yourdomain.com] + play with “date range” advanced search option.

Check who (and what) your site is associated with: search [related:yourdomain.com] to identify your site co-citation.