"Singer" and "Lilly White" are the two basic sewing machine oils. They are the same and are made from mineral oil. Bernina has made a couple different oils with different vescosities (or thickness), but they are basically the same thing -- mineral oil.
You want to avoid 2-in-1 oil. It tends to make a black, sticky mess on a sewing machine.
WD-40 is a solvent, not a lubricant. We use it to remove gummed-up oil, but it has to be completely removed while wet.
Tri-flow is an awesome oil for older machines that tend to get gummed up. It has added Teflon, that gets things moving. Not recommended for newer machines.