My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.
Here is how sitelinks might look for a site:
What are some things that I can do to improve my chances of getting sitelinks?
You will need to have quite a bit of traffic to your site to get those. Keep creating great content. Google will not give those out easy.
Be sure that you have your navigation marked up with semantic HTML and include an id of nav or navigation on your menu. This will help Google know what links are your navigation. (as opposed to a list of links).
Also, once you get them Google will give you an option to control what links you want in your sitelinks via Google Webmaster Tools.
There is no easy way or publicly know algorithm that will give you sitelinks. And I'm sure that algorithm changes all the time.
DO NOTS:
Excerpt from Googler Maile Ohye's explanation on Sitelinks in Google Webmaster Help -
Sitelinks are often produced when there's a high probability that the site is the top match for the user's query. In other words, to have sitelinks ... you'll probably need to be the definitive first result
Sitelinks are completely automated & based on Google's algorithms. To get Sitelinks for a site it helps to have a well-structured website.
Sitelinks are usually taken from the home page. Make sure Google is able to see those links; if your website used heavy JavaScript to construct the DOM it's possible that Google simply doesn't see any link.
To see if this is the case open your website with a command line (no GUI) web browser, such as links (Linux). Alternatively, try to disable JavaScript in your own browser.
You may also maintain a footer that is rich in (text) links; make sure you use good labels. You can think of the footer as a subset of your complete sitemap. This might suggest Google that those links / contents are quite important since they are repeated across the site (assuming all pages have the same footer).
Removal of sitelinks is more straightforward: you can do it via the Google Webmaster Tools.
Google Search Console Help - Sitelinks:
http://support.google.com/webmasters/bin/answer.py?hl=en&answer=47334
Demote a sitelink URL:
On the Search Console Home page (formerly "Webmaster Tools"), click the site you want.
Under Search Appearence, click Site Links.
In the For this search result box, complete the URL for which you don't want a specific sitelink URL to appear.
In the Demote this sitelink URL box, complete the URL of the sitelink you want to demote.
When the website schema is used, it can help Google resolve the ambiguity of which links on a site are the main sub-site links.
{
"@context": "https://schema.org/",
"@type": "WebSite",
"name": "example",
"url": "https://example.com/",
"hasPart" : [
{
"@type": "WebPage",
"name" : "About",
"url" : "https://example.com/about/"
},
{
"@type": "WebPage",
"name" : "Another Part",
"url" : "https://example.com/another/"
},
{
"@type": "WebPage",
"name" : "Google gives three site links today",
"url" : "https://example.com/last-but-not-least/"
},
{
"@type": "WebPage",
"name" : "Buy now, as determined keyword intent",
"url" : "https://example.com/by-intent/"
}
]
},
Depending on the site's structure, there may be no unambiguous indications of which pages should be considered the site links.
Note as of today the site links are shown as links directly under the description. Not the full panel: Google is in the process of cleaning up or simplifying the look of the SERP.
