What are the Most Dissimilar MNIST Digits?
Not sure if this constitutes a &quot;study&quot; but I have investigated using PCA to decompose the MNIST dataset to visualize in 2D:

pca = PCA(n_components=2)
pca.fit(ziptrain[:, 1:])
Z_train = pca.transform(ziptrain[:, 1:])

fig, ax = plt.subplots()
for digit in np.unique(ziptrain[:, 0]):
    x = Z_train[ziptrain[:, 0]==digit, 0]
    y = Z_train[ziptrain[:, 0]==digit, 1]
    ax.scatter(x, y, label=int(digit),
               alpha=1.0, edgecolors='none', marker='${}$'.format(int(digit)))

ax.legend()

This yielded the following plot:

Visually I would say that 1's and 0's are very different. I'm sure you could continue the study by clustering them &amp; measuring the cartesian distance from the centroids of their respective clusters.
It is important to note though that your definition of what &quot;different&quot; means will change the outcome of such an investigation and that this is simply one approach.