Covers are allowed because they are performed in venues with ASCAP, BMI, and SESAC licenses - http://www.ascap.com/licensing/licensingfaq.aspx. Covers performed in public in a venue without one of these licenses are not explicitly allowed.
Michael Jackson made billions off the Beatles catalog, for example. The songwriting copyrights, not the recordings.
"Since the Copyright Act of 1909, United States musicians have had the right to record a version of someone else's previously recorded and released tune, whether it's music alone or music with lyrics.[7] A license can be negotiated between representatives of the interpreting artist and the copyright holder, or recording published tunes can fall under a mechanical license whereby the recording artist pays a standard royalty to the original author/copyright holder through an organization such as the Harry Fox Agency, and is safe under copyright law even if they do not have any permission from the original author. Other agents who can facilitate clearance include Limelight, the online mechanical licensing utility powered by RightsFlow. The U.S. Congress introduced the mechanical license to head off an attempt by the Aeolian Company to monopolize the piano roll market.[8]"
[1] https://en.wikipedia.org/wiki/Compulsory_license#United_Stat...
Any artists who haven't allowed their recordings to make it to electronic distribution are also drowned in poor covers online. AC/DC was a great example of this problem until they finally gave in and released their stuff digitally. There's still a ton of AC/DC cover recordings on iTunes.