One way to do it is to force some extra space below the plots. Then you can fit the legend right there and have one "global" legend.
import matplotlib.pyplot as plt
import numpy as np
plt.close('all')
fig, axlist = plt.subplots(3, 3)
for ax in axlist.flatten():
line1, = ax.plot(np.random.random(100), label='data1')
line2, = ax.plot(np.random.random(100), label='data2')
line3, = ax.plot(np.random.random(100), 'o', label='data3')
fig.subplots_adjust(top=0.9, left=0.1, right=0.9, bottom=0.12) # create some space below the plots by increasing the bottom-value
axlist.flatten()[-2].legend(loc='upper center', bbox_to_anchor=(0.5, -0.12), ncol=3)
# it would of course be better with a nicer handle to the middle-bottom axis object, but since I know it is the second last one in my 3 x 3 grid...
fig.show()
Now there will be an label below the second last (bottom middle) axis area, thanks to the bbox_to_anchor=(x, y)
with a negative y-value. Depending on how many different subplots you have, and how many different lines you plot in each subplot it might be better to keep proper track of the different line-objects. Maybe append them to a list.
For me it the output figure looks like

Does this give you want you were looking for?