Hackamores and other bitless bridles are not permitted in the hunter or equitation rings, so you may only use them in jumper classes.
Hackamores are generally not used with nosebands because of their location on the horse's face.
Some horses are extremely sensitive to the pressure exerted by a hackamore while others can ignore it quite easily. Be careful with these less sensitive horses as it is all too easy to maintain a heavy pressure on the nose when riding them, and this can restrict breathing or bruise the nose. Adjust the hackamore as high as you can to avoid restricting breathing or damaging cartilage (basically put it in the highest location at which your horse will still listen to it).
There are two types of hackamore that are seen in the jumper ring:
Jumping hackamore - The jumping hackamore looks quite similar to a regular cavesson noseband. It is made of very stiff leather and has attachment rings for the reins right on the cavesson. There are no shanks to create leverage, so in the right hands it is very mild.
Mechanical hackamore - The mechanical hackamore differs from the jumping hackamore in that it has shanks to produce leverage. The piece over the nose can be made of various materials (the material will, in part, determine the severity) and the curb strap can be made of either metal chain or leather (leather is milder). The length of the shanks will also alter the severity, with longer shanks exerting more leverage for increased stopping power. The hackamore shown here is often called a German hackamore and it's characterised by long thin shanks and a noseband that is usually made of leather or rubber-covered chain (softened by the addition of a sheepskin cover). An English-style mechanical hackamore has shorter, flat shanks and a noseband made of padded leather, usually adjustable in length. Some German hackamores are shaped so that they can be combined with a snaffle for horses that could benefit from both.