When I go in certain shops, and also MacDonalds, in the UK, I very often hear a certain style of record, where a vocalist puts on a certain kind of "whiny", nasal tone to prolong some notes. Also the voice has a certain digitally-processed air. Male and female artists. I believe one of them is called Rihanna. Is there a name for this type of singing?
It's not easy to tell from your question, but I suspect you're primarily talking about "Auto-Tune."
Here is a well-known Rihanna song ("Disturbia") showcasing the technique, and another one ("Live Your Life") that she did with rapper T.I.. It's particularly noticeable when paired with the elongated singing of multiple notes on the same syllable. (That technique is called melisma, its origins are in black American gospel music, and it's been big in pop music for over twenty years now. See this answer for more.)
Auto-Tune started out life as a digital post-production tool designed to subtly correct the pitch of singers who were slightly out of key, and it is still used frequently in that way --leading to the artificial perfection in pitch that characterizes modern pop music. However Cher's hit "Believe" introduced an alternate use of the tool to create an clearly digitized sound similar to a talk-box where the singer's voice slides sharply from one note to the next (in a way that is difficult to duplicate without technological assistance). As popularized by the American rapper/singer T-Pain, this became a pop-music craze that peaked in the latter half of the first decade of the 2000's. It doesn't necessarily define one specific genre or sub-genre, but it's most associated with 2000s R&B, rap and pop.
As far as the "whiny" nasal sound, that's just the default vocal style of current Top 40s pop. Nasal singing is often considered a defect, particularly in the European classical tradition, but there are many traditions where nasal singing is preferred: Female Japanese singers frequently affect a nasal sound to sound younger, and some styles of Mexican music pair two singers, one with a nasal sound and one with a non-nasal sound. Nasal singing is preferred in some African cultures, and is also often heard in Broadway musicals (because it is naturally louder and easier to hear/understand). It goes in and out of style in pop music as well --for instance, it was big in 1920s and 30s American pop. It's perhaps particularly popular among current teenagers because it tends to grate on the ears of older listeners.