Laser speckles commonly demonstrate Rayleigh intensity statistics and only possess short-range correlations. Recently we develop a method of customizing the intensity statistics of speckle patterns and introducing long-range spatial correlations among the speckle grains. The tailored speckle patterns exhibit radically different topologies and varying degrees of spatial order. The various families of speckles are created by encoding high-order correlations into the phase front of a monochromatic laser beam with a spatial light modulator. This work provides a versatile framework for creating complex light fields and controlling their statistical properties for varied applications in microscopy, imaging, and optical manipulation. As an example, we design and create special speckle patterns for parallelized nonlinear pattern-illumination microscopy based on fluorescence photoswitching. In a proof-of-principle experimental demonstration, we obtain a spatial resolution three times higher than the diffraction limit of the illumination optics in our setup. Furthermore, the tailored speckles vastly outperform standard speckles.