At Sunday Vespers, Psalm 114 is sung using this tone; it's the only time I've ever heard it used in the Divine Office. The name, of course, means "Wandering Tone" (literally, "Pilgrim Tone"), and it's fitting to use it to sing Psalm 114, since it describes Israel's flight from Egypt. (At the Vespers I attend, Psalm 115 - "Non nobis, Domine" - is tacked on to 114.) This is also the only tone in which the intonation is sung at every Psalm verse. The tune is very old, according to this page that seems to have something to do with Boston Camerata:
The synagogue gave to the Early Christian church some of its ancient melodies; the recitation formula of the psalm B'tset Yisrael ("When Israel went forth out of Egypt"), for example, survives in the Gregorian chant repertoire as the tonus peregrinus. It is thanks to a Christian that we have the oldest surviving example of written-down Jewish music, the beautiful Eulogy of Moses.
Here is an mp3 of Psalm 114 sung to the Tonus Peregrinus from the Lutheran Liturgical Prayer Brotherhood Psalmody page. The translation is the King James Version; the mp3 includes an antiphon on either side of the Psalm.
1 When Israel went out of Egypt,
the house of Jacob from a people of strange language;
2 Judah was his sanctuary,
and Israel his dominion.
3 The sea saw it, and fled:
Jordan was driven back.
4 The mountains skipped like rams,
and the little hills like lambs.
5 What ailed thee, O thou sea, that thou fleddest?
thou Jordan, that thou wast driven back?
6 ye mountains, that ye skipped like rams;
and ye little hills, like lambs?
7 Tremble, thou earth, at the presence of the Lord,
at the presence of the God of Jacob;
8 which turned the rock into a standing water,
the flint into a fountain of waters.
Here's a bit of explication from the blog Dappled Things:
We're used to most of the Psalms expressing praise or repentance or asking for protection -- things like that. A few of the Psalms are historical, though, and recount some event from the life of the Chosen People. Today's Psalm, the 113th (114th), recounts the going-forth of Israel from barbarous Egypt into the Land of Promise, with the Jordan fleeing before them. This Psalm is very special. In the traditional Roman Rite, this Psalm was sung each Sunday night at Vespers, and it has a special tone that is normally reserved for it alone: the tonus peregrinus, or "pilgrim tone," music well suited to this Psalm of the wandering of God's People in their search for the Holy Land.