My personal preference for webapps is component.
What is your back-end? What is your frontend? Any frameworks? How important is total uglified size to you? i.e. are you working for a company where every 100ms for each 100kb matters? How are you handling other assets like images, css, etc? How familiar are you with node.js and the npm ecosystem?
A few years ago I realized I loved javascript, so I've been studying everything frontend with the hope of eventually leaving the JS hating MS world, and going somewhere more JS oriented. For JS I've learned a good deal of Backbone, Angular, Node.js, and soon I'll be focusing on WebGL.
I am familiar with npm.
I usually roll my own assets. Most solutions I've worked on haven't had a heavy art presence, and having basic-decent photoshop / design understanding has been sufficient.
As for company requirements, I really don't have any. I just know that I'm going to pick at least one on that list to dive into, and was hoping for community input before I put the time to learn investment in a framework. I mean, if most development shops use one over the other, then I'd love to learn that one! Or if one is up and coming, then that would be great too...
Bower really doesn't do much other than resolve dependencies, but that doesn't mean you won't find it useful.
If you are really into the node.js ecosystem and your back-end is node.js, then browserify is worth checking out. If not, I'd opt for component.
The reason bower is so popular is because it doesn't do very much.