A list of datasets categorised on ML problem type.

List of easily accessible datasets for MR and CT (brain) images along with the data format in parenthesis at the end of the bullet:

Brainweb: Simulated normal and MS brains with tissue/lesion segmentations (MINC)
Kirby 21: Set of 21 healthy patients scanned twice (NIfTI)
IXI dataset: Set of 600 healthy subject scans (NIfTI)
Qure.ai CT head scan data: Set of 491 head CT scans with pathology [no segmentation, but radiology report] (DICOM)

Here is a list of not so easy to download (but very useful) datasets.

BraTS 2018 Brain Tumor data: Large set of patients with brain tumors along with the tumor segmentation (mha)
ISBI 2015 Multiple Sclerosis Challenge data: Set of 20 patients with multiple sclerosis (MS) [only 5 have lesion segmentations] (NIfTI)

I have not worked with the set of datasets below, but I know people who have and am including them for completeness.

Classification: Binary and Multi-class

NLP