Paper is an acceptable support for most mediums including oil paint as long as good preparation methods are used. Paper can be made of all kinds of fibers, but the two main materials are wood cellulose and cotton. Paper manufacturing isn’t regulated so there is no standard to judge them by, but the top manufacturers have reputations to keep and will tell you the features of their paper.
For wet mediums like paint, cotton paper is the better choice as most cotton papers are acid free and tend to be thicker than cellulose. Cotton paper also mimics canvas to a certain degree since they are made out of the same thing, and you can expect a similar longevity. The best way to paint with oils on paper would be to mount the paper to a wooden panel, such as hardboard, and then size the paper and cover it with a ground. Watercolor or acrylics can be used directly on the paper, but you will get better results if watercolor paper is used since they have a thin size applied to them already.
Paper that is not acid free shouldn’t be considered for any work that you want to last as non-acid free paper will yellow and become brittle over time. A paper that is acid free and also buffered buys extra insurance since the buffering means the paper is slightly alkaline and will counter any small amounts of acid that come in contact with it.
Obviously, some of the biggest problems with paper as a support is that it can wrinkle, crease, and tear. Storing the paper in flat files will help with this, but mounting paper is probably the best way to prevent the problems. Also, paper will expand and contract with moisture just like any natural material, so for oil paint is has all of the same negatives as a natural cloth support.
Metal has been used for hundreds of years for paintings, but because it is relatively expensive and heavy it doesn’t see a wide use. Paintings of the Old Masters on copper are some of the most well preserved in the world with little or no cracking. Copper and aluminum have similar expansion and contraction coefficients as oil paint and aluminum is nearly identical with acrylic paint.
Some modern supports use sheets of aluminum sandwiched to a plastic core and are very good, but their price is still more expensive than wood, or cloth supports.
Thin metal supports should be avoided as the metal can easily be bent cracking the paint or even putting a permanent crease in the support which would transfer to the painting.
Glass would seem like an ideal support as it is smooth, rigid, doesn’t rot or expand or contract with air born moisture, but it’s fragile nature keeps it from being widely used. Also, some paint won’t stick well to glass and over time the painting could come loose.
As time goes on new supports are coming onto the market with both good an bad characteristics. Unfortunately, there is no way for me to cover all of them so the best way to choose would be to research them yourself based on the particular needs of your chosen medium. A great place to do that is AMIEN.org.