Recently, TIME Magazine had a cover story regarding the Death Penalty in this country.  Included in its coverage of capital punishment in the United States was an opinion piece by sports legend Kareem Abdul-Jabbar.  
Mr. Abdul-Jabbar writes about his arguments for abolishing the Death Penalty.  You can read it here.
This is another example of the growing media coverage of the Death Penalty and the debate over whether or not there should be executions in the United States today.
If you read some of these articles, you may get the idea that the Supreme Court of the United States has the issue of outlawing executions before it.  
The widest reading of things before SCOTUS today would be a wide interpretation of the lethal injection method of execution as those issues have been presented in an Oklahoma case, Glossip v. Gross.  It’s debatable if that case is really arguing against lethal injections as a method of execution or it is is targeting the use of a particular chemical in an execution using the lethal injection method.
And remember, there are several alternative methods of execution with SCOTUS approval already in place if the lethal injection method of execution were to be ruled unconstitutional.