TITLE: Consistency Strength of the Failure of Square on Singular Cardinals
QUESTION [5 upvotes]: Q1. What is the consistency strength of the failure of square on singular cardinals?
Q2. What are known as partial results in this direction?

REPLY [4 votes]: One place to start reading is this:

James Cummings and Sy-David Friedman, "$\square$ On the Singular Cardinals". The Journal of Symbolic Logic Vol. 73, No. 4 (Dec., 2008), pp. 1307-1314.