Does the Shappening mean HTTPS can be broken?
SHA1 certificate is getting weaker because computing power grows to Moore's Law. According to Jesse Walker, who made a back of the envelope calculation of the cost of SHA1 collision attack: 


  the cost of the attack will be approximately:
  
  
  \$2.77M in 2012
  \$700K by 2015
  \$173K by 2018
  \$43K by 2021
  
  
  A collision attack is therefore well within the range of what an organized crime syndicate can practically budget by 2018, and a university research project by 2021.


To your question:


  What are the practical 'something bad's?


The practical "something bad" is that sometime this year, all major browser vendors (Google, Firefox, Microsoft) would distrust sha1 certificate. This means that your site/servers will be blocked with a big security warning, it might actually be more sensible to not use HTTPS at all than to use SHA1 certificate at that point. Some browsers now already no longer shows the secure lock icon on sites with SHA1 certificate.


  Once an attacker has the capability to create a SHA-1 collision, what then can they do?


Certificate signature is made by taking a hash of the public key and attached data (owner identity and Common Name/SAN) and then the CA signing that hash with the CA's private key.

If you generate two keypairs with the same SHA1 signature, you can convince a CA to sign one of the public key and that signature will also be valid for the other key. From there on, it's a simple matter to implant the CA's signature on the second certificate. That second certificate could be for a different domain that you don't actually control.