I currently use a DataTable to get results from a database which I can use in my code.
However, many example on the web show using a DataSet instead and accessing the table(s) through the collections method.
Is there any advantage, performance wise or otherwise, of using DataSets or DataTables as a storage method for SQL results?
It really depends on the sort of data you're bringing back. Since a DataSet is (in effect) just a collection of DataTable objects, you can return multiple distinct sets of data into a single, and therefore more manageable, object.
Performance-wise, you're more likely to get inefficiency from unoptimized queries than from the "wrong" choice of .NET construct. At least, that's been my experience.