Yes, you can use SDS-PAGE in some cases, like, separation of very small fragments of DNA usually less than 1kb. Here, SDS-PAGE can provide you a better resolution. As SDS- PAGE is used for separation of small molecules (Proteins) and agarose gel electrophoresis is used for separation of large molecules (DNA) due to porosity differences in both gels. Anyway, if you use SDS-PAGE for DNA analysis you should use ethidium bromide staining rather than commasive blue staining.
Usually, in DNA gel electrophoresis urea or formamide is used to break apart the hydrogen bonds that join the chains of nucleic acids. These compounds accomplish this by forming their own hydrogen bonds with the nucleotide bases. Once linearized in this manner, the (single-stranded) DNA has a charge proportional to its mass as it is uniformly negative throughout. Therefore it may be run on a charge gradient to determine its size.
DNA gel electrophoresis is usually done through an agarose gel. The higher the percentage of agarose, the smaller the fragments are that can be detected. Nucleic acids are bigger than proteins and a polyacrylamide gel (as used in protein SDS PAGE) may be used for very small fragments of DNA (under 0.2 kB). This is because very high percentage agarose gels (such as 3%) do not set evenly. Traditional sequencing (Maxam-Gilbert or Sanger) was conducted with PAGE and could determine DNA size difference down to a single nucleotide.
So, yes: you can conduct DNA gel electrophoresis using a polyacrylamide gel.
However, whether you may use SDS in your experiment is another story. Sodium dodecyl sulfate (SDS) is an organic compound used to denature proteins prior to PAGE. It also may be in your shampoo. Added to a cell lysate, SDS (a surfactant) coats the denatured proteins and carries a strong negative charge, allowing proteins to be separated by size via a charge gradient. DNA already has a negative charge, so treating with SDS is not necessary.
As Nazmul said, it did have an application. Nowadays, the use of SDS-PAGE can be supplemented with capillary electrophoresis. Making the polyacrylimide gel is a bit more tricky than agarose and has a lower shelf life.