While I had a very bitter experience with MQ Series partially due to the fact that it was pushed on to us (a Microsoft shop) by the partnering company, the use of MQ Series (Or any Messaging system) was an integral part to the application.
Essentially we were building a process that handled supply chain fullfilment for backorder items. If our partner a distributor didn't have the items their customers wanted, they would send a message into a B2B site, that would target potential companies that could fulfil the order.
We had built two different flavors of integration. The first was an ftp approach where fixed width files were sent back and forth at regular intervals, and we had added all sorts of rules to help ensure we didn't miss any data.
The second was using MQ Series where the messages were placed into a queue using guarenteed delivery. Then we would pop the queue and process the messages. The queing system was great benefit here as it allowed us a reliable way to transmit critical messages that resulted in real money being moved around.
On the flip side with the same MQ Series we had to implement a synchronous query to get information. We wanted it to be synchronous because our users accessing this via the web would wait to get the information. Doing this over MQ Series was a very interesting and painful challenge. The only reason MQ was used here was because it was an existing line of communication and the query functionality already existed.
A second example and this time was using MSMQ was a site that collected information from dialhome code injected into client applications. The dialhome code would collect feature usage statistics like Microsoft's SQM program. When the messages came in to the web service we would drop them on a queue, Then we could have any number of application servers popping the messages and pushing them to the database to be rolled into the warehouse.
MSMQ here ensured we could handle bursts of messages by quickly placing them on the queue. This help the scalability and reliability of the system.