TL:DR; Mongo works for me ('us'), and when I get some time I'll write a post on what we're doing with it, and why it works for us.

I am ('we are') using Mongo for a public transit planner in South Africa. It's not yet production-ready, but beta testing is going well.

Let me paint the picture before I go on to justify our use of Mongo.

In South Africa there are trains, buses, minibus and other services (metered taxis, shuttles). Trains run on stop-by-stop schedules, all of the bus services just a normal departure-based schedule, and minibus completely dynamic. In order to implement a well-organised integrated planner, you have to view all of these as one 'type' of service.
There's then how pricing is calculated for each of the services. There are many different ways in which pricing is calculated, being: (distance-based, zone-based, pricing matrix, fixed minimum with variable charge etc.), then there's ticketing, discounts etc. That too we needed to represent in a simple structure.

Now, my SQL is pretty good, I don't frown upon indices nor joins, but what I can say is that in my initial implementation of the whole idea, I faced a number of problems, being:

(1) what level of normalisation/denormalisation is necessary? (i.e. what should I join, what should I keep in the same table)

(2) I'm essentially using a graph, except that nodes aren't always connected, so how do I traverse the graph when it's actually 'broken'? (excuse me if I get the terms wrong, I'm actually a technical accountant, programming is my second love)

(3) I'm working with location data, I can't expect to use the Haversine formula or equivalents, how can I index both locations, and routes? How do I even store routes? (blob of serialised arrays?)

(4) How do I reduce development time, to reduce the amount of time I spend refactoring schemas/code when I want to implement some new 'shiny' feature?

These were my main concerns, as they were the problems that I had with MySQL.
PostgreSQL would have done a good job for (1) and (3), and maybe (2), but I was still concerned with (4) as working with PHP/MySQL isn't the friendliest of things. Doing 'in()' queries is one example, as I have to parse an array to a string before using an in() function.

Mongo initially appealed to me because it was marketed as 'schemaless', but even someone with little knowledge as I knew that it should be taken with a grain of salt. The benefit here was that I could store different services with different attributes in one collection. If a service is a minibus, I add all the fields that I need for the minibus, and omit the ones for a train for example. Similarly with the pricing structures.

At first it was difficult grasping the 'store everything in one document' model, but I got the hang of it, and now my schema is 'frozen', so (1) has been solved. I don't use joins or any simulation thereof. Yes, because storing I store what I need in one document, I don't need to go back to Mongo to find joining data.

(2) A graph database wouldn't work well for me, because even though services link together, there are instances where the commuter will have to walk to join another service. How do I do that? I initially created manual walking links in MySQL but that was naive and stupid (trying to avoid Haversine). Obviously PostgreSQL distance-based queries would also work here.
Another thing against a graph database is that my project doesn't just rely on traversing graphs all day, there are other things which I need to do, like analytics.

(3) To be honest, even though I'm confident with my SQL knowledge, all the PostgreS/PostGIS functions felt a bit intimidating at first. I can't afford a $250k/year DBA, so I have to know what I'm doing on the database as well as the client/server.
I find learning how to use Mongo to be easy, even though people say their query 'language' is (insert bad word), I find it quite user-friendly.
Mongo's geospatial support, and GeoJSON as of 2.4, made things like storing a route, and running typical queries on it, very easy.

One other thing that we discount is the transparency of data in Mongo, yes they should use field compression and other space-saving techniques, but until them, I see the benefit in being able to do a db.collection.find() and being able to look at the whole document without having to join any tables (it's just a bit quicker I guess).
Which brings me to (4): the most fun thing that I had to implement was scheduling, bearing in mind that there are scheduled services, and variable/random ones. Let's say I have a separate table for schedules, and I want to find:
- x services,
- that start/pass at [lat,lon]
- which are operating during ab:cd AM
- which have the next schedule/estimate within m and n.

Sure, you could do it with a JOIN, but why do that when you can do it all from a single document?

Lastly, experienced programmers tend to take for granted the benefit of simplifying certain things for novices/beginners. The reason why JSON is taking over, besides that it's a more compact and readable expression than XML, is that it's also easy to work with. Why should I worry about converting associative arrays to strings in order to do an in()? Instead of saying something like "in(array)" directly?
Even though I was forward thinking in my schema design, there were a few changes along the road when I realised that something wasn't working. Making a change in the schema was quick, and I didn't have to spend a lot of time making sure that my data is still fine.

Please note that I didn't talk about 'web scale, speed' or all those other things. Under my current hardware I would need to cover 3 countries' transit systems before I need to shard. I am running a single-node replica so I can enable backups.

That's just my view, I wrote this in pieces, so I might appear to be all over the place. I'll write a thought-out post detailing why Mongo is currently working for us/me.