Ariel's song performed by a Celtic musician. The tone is mournful, and the various voices floating in near the end suggest the kind of magic inherent in the script. In addition to attaching the song to images of water, the instrumentation suggests the sea-shanties of yore, giving the song a specifically nautical context which connects with the text. The sound is not modern, but neither is it attempting to be historically accurate -- the important element seems to be the tone and the motif of the sea, rather than the play or the fact that it is Shakespeare.
A low quality video of a single performer doing a rendition of Ariel's song as part of a vocal concert, resulting in a Song with a stiff, classical feel. This performance seems to suggest the treatment of the song as a piece of literature, important as an intellectual status symbol, rather than a part of the dramatic action and aesthetic of a play.
A modern adaptation by a single musician, which taps into the often playful nature of the spirits on Prospero's island. This rendition has some scariness to it, which reflects the strangeness of hearing invisible voices on a Mediterranean island. By making no allusion to the classical in the music, the artist transports the poem to a modern context, rather than putting it on a literature pedestal
A particularly interesting recording of Ariel's Song put together by a group of elementary school children can be found here through the BBC. The main goal in this rendition seems to be understanding the song as it functions in the play text -- to disorient and scare Ferdinand, and conjure the images of his drowned father in terrifying, haunting ways.