How do you just show the text label in plot legend? (e.g. remove a label's line in the legend)

You can just set the handletextpad and handlelength in the legend via the legend_handler as shown below:

import matplotlib.pyplot as plt
import numpy as np
# Plot up a generic set of lines
x = np.arange( 3 )
for i in x:
    plt.plot( i*x, x, label='label'+str(i), lw=5 )
# Add a legend 
# (with a negative gap between line and text, and set "handle" (line) length to 0)
legend = plt.legend(handletextpad=-2.0, handlelength=0)

Detail on handletextpad and handlelength is in documentation (linked here, & copied below):

handletextpad : float or None

The pad between the legend handle and text. Measured in font-size units. Default is None, which will take the value from rcParams["legend.handletextpad"].

handlelength : float or None

The length of the legend handles. Measured in font-size units. Default is None, which will take the value from rcParams["legend.handlelength"].

With the above code:

With a few extra lines the labels can have the same color as their line. just use .set_color() via legend.get_texts().

# Now color the legend labels the same as the lines
color_l = ['blue', 'orange', 'green']
for n, text in enumerate( legend.texts ):
    print( n, text)
    text.set_color( color_l[n] )

Just calling plt.legend() gives:

At that point, it's arguably easier to just use annotate.

For example:

import numpy as np
import matplotlib.pyplot as plt

data = np.random.normal(0, 1, 1000).cumsum()

fig, ax = plt.subplots()
ax.plot(data)
ax.annotate('Label', xy=(-12, -12), xycoords='axes points',
            size=14, ha='right', va='top',
            bbox=dict(boxstyle='round', fc='w'))
plt.show()

However, if you did want to use legend, here's how you'd do it. You'll need to explicitly hide the legend handles in addition to setting their size to 0 and removing their padding.

import numpy as np
import matplotlib.pyplot as plt

data = np.random.normal(0, 1, 1000).cumsum()

fig, ax = plt.subplots()
ax.plot(data, label='Label')

leg = ax.legend(handlelength=0, handletextpad=0, fancybox=True)
for item in leg.legendHandles:
    item.set_visible(False)
plt.show()

I found another much simpler solution - simply set the scale of the marker to zero in the legend properties:

plt.legend(markerscale=0)

This is particularly useful in scatter plots, when you don't want the marker to be visually mistaken for a true data point (or even outlier!).