A conjecture of Katz and Sarnak predicts that the distribution of spacings between ``straightened” Hecke angles (corresponding to Fourier coefficients of Hecke newforms) matches that of a uniformly distributed, random sequence in the unit interval. This comparison is made with the help of local spacing statistics, such as the level spacing distribution and various types of correlations of the Hecke angles. In previous joint work with Baskar Balasubramanyam and ongoing joint work with my PhD student Jewel Mahajan, we have provided evidence in favour of this conjecture, by showing that the pair correlation function of the Hecke angles, averaged over families of Hecke newforms, is expected to be Poissonnian, with variance converging to zero as we take larger and larger families. In this talk, we will explore various types of questions arising in the study of the local behaviour of sequences of Hecke angles, and explain the above-mentioned results.