What should base kitchen cabinets sit on top of? Why? Why not?
Always an interesting choice. Most new construction puts cabinets on the subfloor. One of the reasons for doing this is to avoid damaging the new floors during construction. Floors are one of the last items to complete so it is much easier to install cabinets first.  The downside is that when a future remodeling is done, the footprint of the old cabinets rarely matches the new plan. Now you are faced with stitching gaps in the floor, matching color, materialt etc. The other consideration is that if the flooring is thick, like hardwood or ceramic tile, it locks the base cabinets in place and they can be very hard to remove if for some reason one needs repair or replacement. This method is most common however.
The second option is to lay the floor first and install the base cabinets on top of the finished floor. I personally like this method if using a very long lasting durable floor like hardwood or upscale tile. I would not do this over vinyl or cheap flooring that will not outlast the cabinets.  For a DIYer, this method can be much easier. There is much less detailed cutting and fitting around cabinets, the fit to the walls does not need to be perfect where it is hidden by the cabinets, and less trim may be required along kick panels. If the footprint of the cabinet layout is complicated with lots of different depths, or if islands are involved, this method is popular.

So, there is really no right or wrong. The decision depends on the kind of flooring you select, grade of cabinets, and long range expectations. 
Side note: Kitchen cabinets are one of the most important fixtures in your home. They get a lot of use and abuse. A good investment is to purchase the best quality your budget can handle. I am not a fan of IKEA products. They do offer lots of bells and whistles for budget minded consumers, but the intrinsic quality does not compare to even mid grade cabinets constructed with hardwood frames, five piece boxes etc,  fastened together with glue and screws and have quality drawer slides and door hinges etc.