The first order of business is to understand what the oils do.  The first thing is they lubricate the pump.  Because the tolerances in these pumps is pretty tight it is important for the mating surfaces to receive proper lubrication.  The correct viscosity for the speed and size of the pump is important.  The second thing the oil accomplishes is to cool the pump.  The majority of these types of vacuum pumps are air cooled so the oil is the "heat exchanger" that carries the heat from the pump itself to the oil sump of the pump where the heat is dissipated.  The third and most important consideration is as a vacuum seal internal to the pump.  The oil involved is intimately involved with creating and maintaining vacuum pressures in the pump and the system.

Basic vacuum pump oils

Acid treated vacuum pump oils - We have one of these we have found we call "Crystal White".  It has fantastic vapor pressure numbers and like 10-8 or 10-9 (so stated) is water white with all the nasty sulfur and such stripped out.  The problem is we have a hard time getting it.  We can get it, it just takes some extra effort and patience. 

Perflorinated oils - Brand Names are Fomblin by Montiedison, Krytox by DuPont - These oils are often used because they are chemically inert or "un-reactive" with other chemicals.  They are incredibly expensive, heavy molecular weight and sometimes actually degrade what ultimate vacuum the pump can obtain but if you gots to have it you gots to have it applications.

Phosphate Esters - Has been used in oxygen rich systems in the past but no company around will sell you this material and certify that anymore.

Silicones and Derivatives - Not very good "lubricants"

Synthetics - There are so many....