The geometric Jacobian provides all the information you need for singularity or manipulability analysis. Linearly dependent columns correspond to joints with parallel axes. More information about Jacobians for under-actuated manipulators (as is your case) can be found in my book Robotics, Vision & Control" section 8.4.1.
For information about computing the "screw based Jacobian" check out the book Modern Robotics by Park&Lynch.
Adding to Peter Corke's answer, there's also a Coursera course by Kevin Lynch which uses the Modern Robotics book as a reference and explains how to derive the screw based Jacobian. The Jacobian can be either with respect to the "space frame" (frame attached to base of the manipulator) or the "body frame" (frame attached to the end-effector). Here's a YouTube video from that course that explains the derivation.