I have kept more than a few species of mantises now since I first started raising them back in 2009. Some species have been more satisfying to own and photograph than others.
European Praying Mantis
This was the first species I came across and raised. While working at outdoors amusement park in Colorado I came across female one. After looking at it I left her alone but few a minutes later I came across male of the same species. I assumed they were the same species from the same black spot on inside of their arms and I knew that some insects were sexually dimorphic. So I decided to capture them. I cut the top off of two pop bottles, popped them inside them, and inverted the top back onto the base in the classic DYI wasp trap and took the home. They both had wings so I knew they were adults. I bred them and classically the male was eaten. This would be the genesis for the next set of mantises I photographed the following spring.
The species is fair sized and easy to raise. They have the classic mantis behaviors of cat-like cleaning, and ambush preying. They do have a habit of just hanging from the top of cage and do not really roam unless hungry or searching for a place to lay eggs.
They accept pretty much anything you put in their cage and will accept uncooked meat with the touch mouth method without difficulty. Best of all since they are wide spread in the US you can release them if you breed too many. They are also cheap available both from garden stores and breeders.
Photographically they are about the perfect size. Large enough to be able work with easily and not have to go beyond 1:1 while still being small enough to blur out the backgrounds. Their pseudo-pupils (black dots on their eyes) are almost always easily apparent in images. They still bend and the abdomen thorax joint after they've gotten their wings allowing them to 'pose' even at adult. They will pose for awhile but will start actively running about after 10-20 minutes of shooting.
Chinese Mantis
I also caught this species wild in Nebraska. They are common and easy to find due to their numbers, size, and appearance near lights at stores at dusk. One of the largest mantis species they can be little intimidating to work with, especially once they have their wings. Unlike other species I've had, it did not seem particularly prone to flying into your face. This was the easiest species to care for. You can feed them just about anything and they will take it with aggression. While they do the normal wait and ambush, with lively prey they will charge it. This means they will actively eat each other, more so than other species I've kept. Like the European they just kind hang out in the lid of the cage.
Photographically they are almost too big. 1:1 is not necessary for most shots and you will need larger shooting areas to hide edges and backgrounds. However, because of their size it can be easier to put whole mantis into focus compared when working with smaller mantises. Similar to European, they bend at the abdomen and thorax as adults and they will break and run after 10-20 minutes of shooting.
Ghost Mantis
First mantis I ordered off the web. Best way to describe this mantis is ultra-chill. Pretty much wait and catch only, no charging down their food. Will stick pretty much to the roof of cage like most species. Takes crickets well and will eat insects that seem much too big for such a small mantis. It isn't food shy at all. After it outgrows fruit flies the mantis is pretty idiot proof. I did at times feed it meat but it was much harder to get it to eat than the previous two species. I've only kept one and I never bother to sex it. One huge plus for this species is that you can raise them as a group and can pretty safely put them in group shots, or at least so says the web.
This is one the easiest species of shot. It loves to pose and will hold poses for minutes at a time and will rarely break and run even when prodded into different positions. As an adult it never took flight when being worked with. It does have two draw backs, its psuedo-pupils are often not easily discerned from the rest of the eye and as adults don't like to bed. They will rear up at least, but not to the extent of the European or Chinese mantis. They are also much smaller than the previous species, prohibitively so when young. Expect to need a bellows or tubes if you want to work with the species beyond full body shots when young.
Budwing Mantis
This species was kind of lack luster to me. It lacked the exoticism of the ghost mantis and was not free like the European or Chinese mantis. Behavior wise it was pretty inactive. Sat on the roof cage, no roaming, wouldn't look at you. Just kind of generic without being free.
Photo wise it was okay until reaching adult. Pseudo-pupils easily apparent. Apealing face and very defined palps make for great up close portraits but otherwise much the same as European or Chinese mantis. Unlike them however once it has its wings it becomes mostly useless. Most of the time it will lay parallel to the surface in a perfect straight line making unsatisfying and difficult to photograph. Half the appeal of these insects is that they stand up something like we do. This issue might only be with the male of the species. Per the name the females have much smaller and less restrictive wings which may allow them to rear up for better posing. I did enjoy that eyes were a pale lavender which easily set them apart from the rest of the face.