New Year’s Eve is thought of first s a time of revelry. And of course, the planning of resolutions.
114. Popular piety has given rise to many pious exercises connected with 31 December. In many parts of the Western world the end of the civil year is celebrated on this day. This anniversary affords an opportunity for the faithful to reflect on “the mystery of time”, which passes quickly and inexorably. Such should give rise to a dual feeling: of penance and sorrow for the sins committed during the year and for the lost occasions of grace; and of thanks to God for the graces and blessings He has given during the past year.
Gratitude and penance: good thoughts, to be sure. But not frequent these days, I would think.
These sentiments have given rise to two pious exercises: prolonged exposition of the Blessed Sacrament, which afford an opportunity for the faithful and many religious communities for silent prayer; and the singing of the Te Deum as an act of community praise and thanksgiving to God for the graces received from Him as the year draws to a close(Cf. EI, Aliae concessiones, 26, p. 71).
The Te Deum … is most well known in its paraphrase, “Holy God, We Praise Thy Name.”
In some places, especially in monasteries and in associations of the faithful with a particular devotion to the Holy Eucharist, 31 December is marked by a vigil of prayer which concludes with the celebration of the Holy Mass. Such vigils are to be encouraged and should be celebrated in harmony with the liturgical content of the Christmas Octave, and not merely as a reaction to the thoughtless dissipation with which society celebrates the passage from one year to another, but as a vigil offering of the new year to the Lord.
The only time I’ve participated in such a vigil was at the very end of 1999. My diocese suggested Eucharistic adoration, but the parish decided to have a family night, late into the evening, then a Mass at 11pm. Post-Communion was when the new millennium was welcomed in.
The Directory on Popular Piety and the Liturgy is online at the Vatican site.
Catholic Sensibility 
In many African and Afro-caribbean communities the keeping of vigil at the New Year is deeply popular as an expression of popular piety. As an Anglican, I note that most of the C of E churches in our Deanery having Exposition of the Blessed Sacrament or a Mass similar to that of Christmas Midnight. They tend to be far better attended than Christmas services! Being Anglican, our parish has an 11.30pm Votive of the Holy Name, which is still kept by us on 1st January.