When members of the Swedish royal family travel, they use normal scheduled flights with basically one exception: official state vistits, when they use a government jet (a Gulfstream) which is painted in the colours of the Swedish air force.

I have a friend who works for the VIP service at ARN and he has told me that the royal family usually travels with SAS, although not always. One time the King actually travelled on Sterling from BLQ to ARN, probarbly because Sterling is the only airline which flies nonstop BLQ-ARN and time was more important than comfort...