As someone who shoots out-of-doors a lot, I am very pleased to have a self-cleaning sensor. I don't see this feature mentioned much in camera reviews, and I am wondering whether they are now a standard feature on DSLRs.
Among digital cameras with interchangeable lenses, this is now an almost standard feature.
Let's look at DSLRs first, since you mention those explicitly. Among 105 DSLRs released since May 2005, only 7 do not have automatic sensor cleaning. Half of those are from 2010 and earlier with the latest two not having sensor cleaning being the Sigma SD1 Merrill from 2012 and Nikon D3400 that was released in 2016.
For mirrorless cameras, integrating cleaning is also now the norm. Among 123 mirrorless cameras launched since May 2005, only 10 do not have integrated sensor cleaning. Oddly, unlike with DSLRs, the number of mirrorless cameras without sensor cleaning has not been diminishing. In 2013, there were 2, 3 in 2014, 1 in 2015 and 4 (!!!) in 2016. That latest 2016 batch is the Sigma SD Quattro, the Sigma SD Quattro H, the Hasselblad X1D-50c and the Canon EOS M5.
May 2005 is when I launched Neocamera, so its database includes all cameras launched from then on by all major manufacturers. You can use the Camera Search tool to find out which camera has built-in Dust-Reduction by selecting the corresponding icon.