The Vigil of the Epiphany is a semi-double of the second class. It is a also a priviliged Vigil of the second class.
The Vigil of the Epiphany takes the place of the Office of the Sunday which occurs from the 1st to 5th January and has all the privileges of a Sunday both in concurrence and occurence. (Rubric in the Breviary.) At Mattins the Office is as on the Circumcision except the lessons that are proper. Mattins has three nocturns and nine lessons. At Lauds and the Hours the antiphons are those from the Circumcision, the antiphon on the Benedictus and collect are proper and a commemoration is made of St. Telesephorus Pope & Martyr. At the Hours the psalmody is festal.
At Mass the Gloria is sung, the second collect is of St. Telesephorus, the third collect of the BVM, Deus, qui salutis. The Credo is sung and the preface is of the Nativity.
Vespers are first Vespers of the Epiphany. For the feast and its octave a Doxology in honour of the LORD's manifestation is sung at all hymns of Iambic metre: Jesu, tibi sit gloria, Qui apparuisti Gentibus, Cum Patre, et almo Spiritu, In sempiterna saecula.
In the 'liturgical books of 1962' the Vigil is abolished and the day is another feria of Christmas. Mattins has one nocturn and three lessons. St. Telesphorus is commemorated at Lauds and read Masses. Vespers are the same as the Old Rite.
The Vigil and the Sunday between January 1st snd the Epiphany have become fused in the 1570 missal and its successive editions. The celebration of the Octave Days of the comites Christi feasts as doubles meant that the Sunday, under the 1568-1911 rubrics, was moved until the first free day, the 5th and day of the ancient Vigil. Examining older books one can find two sets of formularies: one for the vacant Sunday with the collect Omnipotens and one for the Vigil with the collect Corda nostra. These, distinct, celebrations were also found in Sarum practice and its closest surviving modern relative the Dominican rite. With the 1911-13 reform reducing the rank of the comites Christi Octave Days the 'real' Sunday had the Feast of the Holy Name transferred from the Second Sunday after the Epiphany to the Sunday before the Epiphany or January 2nd. My understanding is that a Vigil of the Epiphany has been restored in the 2002 edition of the new Roman Missal.
P.S. Having checked the Memoria produced by the 1948 Commission for General Liturgical Reform I noticed that the commission thought '..La vigilia ha semplicemente la liturgia della domenica dope il Natale..' This rather typifies, in the light of the comment by Francis, the whole probelm with committee-work, inorganic, reform of the liturgy. If a Dominican had been a member of the 1948 Commission perhaps things might have been different for the Vigil of the Epiphany at least!