This monograph discusses pointwise Hölder regularity of functions, especially when this regularity is not uniform but varies wildly from point to point. The main tool is the use of two-microlocalization spaces which allow to study how singularities deteriorate in a smooth environment and conversely. Such spaces can be characterized in terms of decay conditions in the Littlewood-Paley decomposition or the continuous wavelet transform. These two-microlocalizations are introduced in Chapter I and as an application, the pointwise singularity of elliptic operators is investigated. In Chapter 2, singularities in Sobolev spaces are discussed. Depending on the type of singularity, the Hausdorff or packing dimension of the set where a certain Hölder type condition holds is estimated. Chapter 3 investigates the relation between wavelet expansions and lacunary trigonometric series. Especially selfsimilarity and very strong oscillatory (chirp-like) behaviour is discussed. Such trigonometric or logarithmic chirps are studied in more detail in the subsequent three chapters. There is for example a simple characterization of such chirps in terms of certain conditions that their wavelet transform should satisfy. The chirp-like behaviour of the Riemann function

$\sum {n}^{-2}sin\pi {n}^{2}x$ is by now well known and is discussed in the last chapter. It has a trigonometric chirp in the rational points

$(2p+1)/(2q+1)$ where

$p$ and

$q$ are integers, and a logarithmic chirp at the quadratic irrationals.