Removing Everything Else
There is a number of types of jewelry that I either don't stock or don't suggest wearing in a piercing. Though I do stock and suggest plugs with o-rings, I'm not a big fan. They tend to be easy to put in and take out but unless they are grooved the o-rings tend to slide off. This can also be a problem in even jewelry with grooves. Removing the jewelry is rather straight forward. Removed the o-ring and then slide the jewelry through the piercing.
What I'm also not a not fan of is flanged jewelry. The reason is that putting the jewelry in and stretching a piercing well beyond it's current size can cause damage to the piercing. Once they are in, they will stay put until the piercing begins to stretch again and then they start falling out. I really strongly suggest doing a soak, compress or at the least taking a hot shower before trying to remove the jewelry. Soaks and compresses should be done with a sea salt and warm water mixture. 1/8 to 1/4 teaspoon per cup or 8oz of warm water. Soak are done by inverting a small cup over the piercing area or submerging the piercing for around 10 minutes. For piercings where is not possible, take a folded up clean paper towel or sterile gauze sponge, soak up the liquid and lay it against the piercing area. Re-submerge the compress when it cools and apply it for roughly 10 minutes. This will make the tissue more flexible and easier to stretch. When removing flanges it's basically force that removes them slowly pushing and turning the plug on its side and then sliding it out.
Nostril bones is another jewelry style that I suggest against. They are a stud with a ball on one side and a charm of some kind on the other. They usually slid right in because the ball is roughly 18g which is the most common size for Nostril piercings. The problem is the post is usually a thinner gauge. The body conforms to the post thickness making removal a problem. In some cases cutting the wire is the only way you can remove them. I really suggest doing a compress as mentioned above before trying to remove the jewelry. This will make the tissue more flexible and easier to stretch. The to remove grab the jewelry on the outside and pull. Hopefully the ball will slide through the piercing. If you are one of the unlucky one that can't get it through, I would suggest cutting the jewelry with wire or tin cutters. Another option would be the inside area of needle nose pliers. Cutting should be a last restore and should only be done if you have no other way to remove the jewelry.