Very nice! I received the notification of this contest via LinkedIn.
I have a couple good examples of when I've used Wireshark to resolve an issue on my network and they're both good but since the rules called for only one I'll just pick one.

About a year or so ago while I was working as a contractor for a major computer processor manufacturing company one of the factory managers contacted me about an issue they were having with a specific tool in the factory. The details at first were sketchy but I did what I always did and started with the basics.
The problem was that each morning around 3:00am this specific tool and only this specific tool would seemingly lose its network connection. It was become very sluggish/slow in passing data and finally just stop entirely with the applications throwing out all kinds of "lost connection" errors.
After tracing out the device to determine the switch port they were connected to I discovered that every morning right around 3:00am there were a lot of out-discards from a few switch ports (which all happened to share the same VLAN) including the port to the tool in question. Not finding any other issues on the switch port or the network in general I decided to see what was going on the wire at 3:00am that might cause this issue. (I suspected it was something with the specific configuration of the tool...)
So one early morning I headed into work and got my work-issued laptop all set up and Wireshark running sniffing the VLAN on that specific switch. I then sat back and just watched as the packets scrolled by. Sure enough, at 2:50am my laptop went absolutely crazy. The amount of traffic coming into Wireshark from the span port was *insane*. I had to pull the network cable from my laptop and even then it took a few minutes for the data which Wireshark had sniffed to save to the hard drive (my work latpop was pretty low-end).
I saved the data I'd sniffed and headed up to my desk. After a few minutes of poking around I discovered what I felt to be the root cause. At 2:50am a production tool on the network (and the same VLAN as the tool having issues) began a *huge* file transfer. I emailed the findings to my customer and they right away figured out the problem.
This huge file transfer was a hard drive image being mirrored as part of a nightly backup. The script controlling the backup was broken and instead of deleting each days backup file it added a new one to the batch each day. So after a week they were mirroring just a ton of data across that VLAN.
They stopped the backup script the next morning and not a single problem. They fixed the script and allowed it to run the next morning and no issues.
That's it! I was quite satisfied that I was able to figure out the cause of the problem and help my customers fix their issues.
Dave