That said, the 10 blue links are becoming less and less important for most users. It is very important if you are searching for some obscure piece of information.
But the vast majority of queries can be broadly categorized as:
1. navigational queries - I'm trying to get to a site and I don't want/know to type the address, so queries such as "fb", "yt" or "bank of america"
2. trending news queries - something is in the news right now and you want to find articles
3. shopping queries - looking for products and doing product research
4. local queries - I want to find the details of that sushi place I like nearby
5. more niche experiences around various events
If you take a look at what Google and Bing are trying to do right now, they are moving away from what was customarily a search engine and more towards being a portal for you to consume all the content directly on their site (stuff like AMP is a good example).
While this is bad for the user on many levels, it does reduce the problem space for companies wanting to create a new search experience.
So the real question is, what are you searching for? if you are looking for a single search engine that can answer all of the above then you only have 2 options (in the US) If you are looking for a search experience that focuses on specific verticals there are many options.
One service I am interested in is Neeva Search https://neeva.co/ which sounds like they are building a more modern search experience which better fits with what users are doing today