I have ran into people on the trails with mules and they LOVED them. One lady ran endurance on her mule...I think her name was Sugar! Nice well behaved mule. Saw a man on a mule up at Ebenezer last month. Again, great looking animal and well behaved.
We rode at Kisatchie National Forest last year and a Mr. Weber was riding a little 3 year old gaited mule. I rode with him twice that weekend and that little mule was THE BOMB. Calm cool and collected and of course tough as a boot!!
I just bought a mule,and I love her! Mules are very sure footed, and I would take my mule out into the mountains any day over my horse. Unlike horses, mule think over a situation rather than react. So it's very rare for a mule to spook, instead they will stop and think it over. You can't force a mule to do anything, you have to teach them and convince them it's a good idea. But mules are very willing and loyal to somebody that they trust. I hope this helps!
I work with 2 mules and personally, they aren't my favorites. It may have just been former training issues, but they're pushy and extremely stubborn. It can be very hard for us to do simple things with them, such as move them over in their stalls or pick up their feet if they don't want to do it.
They are definitely nice and slow and sure footed though. They don't have any "personal space" issues with other horses, aren't spooky, and are generally very safe.
Mules are great! My neighbors growing up had mules, some of which I rode and others I just worked with around the barn. The mare I leased from them even had a mule baby as her first. Just like horses, they all have their own personalities and things that they are better at. I have known mules who were very hot and hard to handle and ones that are very steady, but light and easy. As someone said already, they tend to stop and think about things if they are scared, rather than spook. Once they decide that it is safe they will continue on, but if they are really convinced that there is danger, you are not going to get them to budge. Luckily, they are smarter than horses so you can usually trust their judgment
As far as riding differently, I haven't noticed much difference. I'm sure training is different, but one that is already trained generally responds to the same signals as a horse. I have heard that sometimes they cannot be trained to back however, but I think that depends on the mule.
This is me on one of my neighbor's mules back in high school. Rachel was my favorite mule to ride
I have ridden/helped train a grand total of 3 mules (one draft and 2 QH size). Once they are trained, they are some of the most trustworthy and reliable mounts to be found. Training them, however, can be quite an ordeal unless you've got a ton of experience with them.
If you want a mule, go out and find one that's broke. Once they get to that point, they are really no different than riding a horse as far as a willingness standpoint. Of course, saddle fitting is an issue because most mules have a donkey's flat back and a regular horse saddle will slip and slide all over the place, causing the need for a britchin or a crupper and sometimes a breast collar too.
Be warned though, a good riding mule will often cost 2-3 times as much as his equine counterpart with identical training/temperament, especially if mules are not common to your area.