A preferred solution badge is applied to any solution that is Co-sell Ready. If you get your solution Co-sell Ready you will get the preferred solution badge added in the commercial marketplace.
Featured apps (the dedicated rows at the beginning of the homepages and each category page) are determined a few ways:
1) Earned as a benefit of Marketplace Rewards. So yes, activate your "Marketplace Rewards" benefits in partner center
2) Performance based - all remaining spots after #1 is applied are granted to solutions performing well on customer engagement indicators (high number or monthly growth of clicks, leads, transactions)
The shortest, best answer is: grow the usage of your app through Marketplace!
For VM-based apps, encourage your customers to deploy through the marketplace. As they deploy and grow, so will your presence in the Azure Marketplace.
For "contact me" or trial listings, consider campaigns to drive more traffic to your listing to increase the leads and trials that are being sent or used through your offer.
All Azure Marketplace publishers are able to access category-specific promotions through the go-to-market benefits that are provded once you go live (or if you update an existing offer), so that's a good place to start.
In general, the Azure Marketplace team is measuring the growth of your app in marketplace, traffic to your offer, ratings and reviews, and special consideration for offers that are clear, informative, and include good visuals in the screenshots section.
Please note that featured apps are different from the "preferred solution" icon.
The documentation says: "A Microsoft preferred solution is a cloud application selected for its quality, performance, and ability to address customer needs in a certain industry vertical or solution area. A team of Microsoft experts validates solutions from partners with specific proven competencies and capabilities. These solutions are featured in our cloud marketplace storefronts, Azure Marketplace, and AppSource, as well as in the Azure portal. Preferred solutions on AppSource can be discovered by industry verticals. Preferred solutions on Azure Marketplace and the Azure portal can be discovered across horizontal solution categories.". We have a couple of products in AppSource that have the Preferred Solution tag. We worked with our Partner Development Manager to have the products go through a review process as a part of getting them into AppSource. Hope this helps!