if you were given a preparation of unknown DNA and you wanted to determine if it was linear or circular?

the other answer sucks, it assumes you already know the length of the DNA in the unknown DNA sample. There are LOTS of ways to test whether the DNA is linear or circular or mixed. A few experiments:

1. Treat the DNA sample with an exo-nuclease such as RecBCD. This exo processes both ds and ssDNA.. but it can't touch circular! If the DNA is degraded it's linear, untouched would be consistant with (not prove) circular.

2. Look for supercoiling by running EtBr experiments to unwind the DNA. Only effects circular DNA.

You can run it on an agarose gel. Linear DNA and circular DNA migrate at different rates. Assuming it is a pure DNA sample, you should see it run so far, and you can compare it to a ladder. Say it migrates to the 3000 bp band of a DNA ladder, you can get a plasmid that is circular and 3000 base pairs and compare it to the unknown DNA. If it is circular, it should migrate at the same size. If it is linear, it should migrate higher than the circular DNA. You'll need to use a low percentage DNA gel for this to efficiently separate out band sizes.