Summary: It is proved that the ``canonical proboscis'' domains corresponding to prescribed contact angle \(\gamma_ 0\), introduced in an earlier work by \textit{B. S. Fischer} and \textit{R. Finn} [Z. Anal. Anwend. 12, No. 3, 405- 423 (1993; Zbl 0782.76015)], are critical for the domain and angle, in the sense that (i) a solution of the capillary problem for angle \(\gamma\) in the absence of gravity exists over the domain if and only if \(\gamma\) is closer to \(\pi/2\) than is \(\gamma_ 0\), and (ii) singular behavior at \(\gamma= \gamma_ 0\) occurs precisely over the proboscis portion of the domain. The construction can be effected in a continuum of ways, allowing the proboscis to occupy as large a portion of the domain as desired.