The asymptotic expected running time of quicksort is $\Theta(n \log n)$: this is true for all three pivot methods you mention.
Wikipedia says that the expected number of comparisons is approximately $1.386 n \log n$ when using a random pivot, and approximately $1.188 n \log n$ when using median-of-three pivot. There's some experimental evidence that the number of comparisons might be about $1.094n \log n$ when using a ninther pivot for large arrays, median-of-three for medium-sized arrays, and single element for small arrays. See the following research paper:
Jon L. Bentley, M. Douglas McIlroy, "Engineering a Sort Function". Software Practice and Experience, 23(11):1249-1265, Nov 1993.
(This paper is cited in the Wikipedia article I mentioned above.)
I'm not familiar with the "uniform shuffle" pivot selection method. It sounds equivalent to choosing a random element and using that as the pivot.