Raging Song (Inspired Rage only1) should prevent spellcasting
The argument I will use for this statement is comparing the wording that is on the Bloodrager's Bloodrage ability:
While bloodraging, a bloodrager cannot use any Charisma-, Dexterity-, or Intelligence-based skills (except Acrobatics, Fly, Intimidate, and Ride) or any ability that requires patience or concentration.
Notice the wording is very similar to the Skald's Inspired Rage. Further, there is nothing else on the description of Bloodrage stating that no spellcasting is allowed. Now the Bloodrager, at level 4 gains the Blood Casting ability (emphasis mine):
At 4th level, the bloodrager gains the ability to cast spells even while bloodraging. He can also cast these spells defensively and can make concentration checks for these spells while bloodraging. While bloodraging, he can cast and concentrate on only his bloodrager spells (see below); spells from other classes cannot be cast during this state.
This seems to imply that normally a Bloodrager would be unable to cast during Bloodrage (or this feature would not make much sense).
Since the Bloodrage ability and Inspired Rage use very similar wording one can conclude that the Inspired Rage Raging Song should also not allow spellcasting.
1. Stolen from ShadowKras' answer, who was the one who noticed only Inspired Rage in particular had that effect