Loopshaping is a historically important method for controller design, and is still a good means to train your intuition.

Preliminary lecture slides: loopshaping.pdf

Proofs of Bode gain-phase theorem and phase-area theorem can be found here: bode_relations.pdf

Additional reading

Properties of M- and N-circles are summarized in table 5-1 on page 103 in AD0655861.pdf
The report contains a lot of good stuff!

Bode summarized a lot of his results in the book: http://jontallen.ece.illinois.edu/MISC/Books/Bode-NetworkAnalysisFeedbackAmplifierDesign.pdf The phase/gain formula appears in chapter 14.
See also chapter 7.2 in "Feedback Control Theory", Doyle, Francis & Tannenbaum, 1992

Skogestads and Postlewhite has a nice section (2.6) on loopshaping.

Resonances cause real problems and when companies can't deal with them, they contact Tore. Read Section 5 in 2013cep_PIDfilter.pdf to learn how Tore saved Sandvik about 200 million SEK.
The story is described in Swedish here.

Chapter 14 in Olof Troeng's thesis uses peak filters to actively compensate for a 90kHz ripple indicated in early simulations of the ESS klystron power supply.