If a diskette is physically damaged, you may be able to salvage the data by removing the actual diskette medium from its protective sheath. For a 5.25-inch diskette, take the following steps, which we've used successfully more than once:
Use a razor blade or sharp knife carefully to trim about 1/8 of an inch (3 mm) from the bottom edge of the protective sheath of another diskette that you don't care about. The bottom edge is that nearest the drive door when the diskette is inserted into the drive.
Gently squeeze the two edges adjoining the trimmed edge toward the center to open a gap in the trimmed edge. Grasp the plastic medium and gently slide it out of the sheath, leaving the Tyvek inner liner in place. Discard the medium and save the sheath.
Repeat the first step on the damaged diskette, and then remove the medium, touching it as little as possible. Put your finger in the central hub hole and gently press the medium toward the trimmed edge. If you must grasp the medium itself, do so only at the very edge. Be very careful not to bend or deform the medium while you are extracting it.
Gently slide the medium from the damaged diskette into the new sheath that you previously prepared, making sure that the medium is right-side up, fully inserted, and that the hub hole in the medium is centered in the hub hole on the sheath. Rotate the medium gently by using your finger in the hub hole to ensure that it turns easily.
Insert the patched diskette into the drive and attempt to access it. It's not necessary to tape or otherwise secure the trimmed edge. If you can access some or all of the data on the patched diskette, immediately copy it elsewhere and then discard the diskette.
Salvaging data from a 3.5-inch diskette is usually impossible. Any damage that renders the hard plastic shell unusable normally also destroys the medium that it contains. However, if recovering the data is critical, you can attempt the following steps, which we've had much less luck with:
Locate a 3.5-inch diskette of the same type (DD, HD, or ED) as the damaged diskette that you are willing to destroy. Using a nail file or small screwdriver, gently pry the sliding metal shutter off the diskette and discard the shutter.
Gently pry open the shell. Remove and discard the medium, leaving the two Tyvek inner liners in place. Place the shell aside.
Repeat the first step on the damaged diskette, and then remove the medium, touching it as little as possible. If the medium is visibly damaged, as it probably will be, continuing is likely to be useless. However, we did once succeed in recovering some files from a medium that appeared to be severely damaged, so it may be worth the effort.
Place the medium from the damaged diskette into the new shell you prepared, making sure to orient the medium as it was in the original diskette, and snap the shell closed. Use your fingertip to make sure that the medium turns freely within the shell. Insert the repaired diskette into the drive (you need not replace the shutter first) and attempt to read the diskette.