You got to give us more information here. There are a lot of ways and technologies to build different kinds of web applications.
If by popular you mean widely-used, The LAMP “stack” of Linux, Apache, PHP, and MySQL remains probably the most popular, and certainly the stack in widest use, because of WordPress. All of my customers save two for the last decade use that stack. I have seen Rails and C#/.Net a few times.
I can tell you're in consulting and not in startups...
Yes. My consulting business is taking over failed/stalled projects, usually startups.
For the sake of completion, though I do not personally recommend it, is the MEAN (MongoDB, ExpressJS, A Javascript Framework, NodeJS) stack.
Weebly/Wix are pretty good. Obviously has limits but we’ve built some pretty advanced sites with Weebly (e.g. restaurant site with menu PDF). Don’t have too much insight in regards to underlying technologies but wish you the best of luck.