In general, the Church has encouraged the Charismatic Renewal, provided it is properly grounded in Church teaching and submissive to Church authority. For example, in 1998, Pope John Paul exhorted Catholic Charismatics “to safeguard their Catholic identity” and maintain the proper relationships with their diocesan bishops and the Holy See.
Unfortunately, in decades past, the enthusiasm associated with the Charismatic Renewal often resulted in poorly formed Catholics who left the Church because they perceived their local parishes as “dead,” despite the ongoing reality of the Eucharist and other life-giving sacraments in those parishes. In 2000 and 2001, the Church also warned against abuses in healing services.
Regarding spiritual gifts, the Church makes proper distinctions. The Catechism of the Catholic Church states that “grace is first and foremost the gift of the Spirit who justifies and sanctifies us” (CCC 2003). The Church adds in paragraph 2003 that graces are sacramental graces—gifts proper to the sacraments—or special graces, also called charisms, which in Greek refers to “gratuitous gifts.” These are the graces that the Church calls the extraordinary gifts of prophecy, tongues, or others (1 Cor. 12:4-11) but which are always at the service of sacramental grace and thus the common good of the Church.
If we look at charismatic gifts in this light, we see that the movement of the Spirit works within the Church according to his will and with the cooperation of the faithful. At the same time, the movement recalls the subordination of the charismatic gifts to those gifts given through the sacraments—specifically, the seven gifts of the Holy Spirit, as expressed in Isaiah 11:2-3, given to each Christian in baptism and deepened in confirmation.
You also asked about Charismatic style of worship, including at Masses. The Church does not have specific universal guidelines in this regard, although there are general norms prohibiting the faithful, including priests, to add or change anything in the Mass on their own authority (see Vatican II, Sacrosanctum Concilium 22.1-3).
On the other hand, Charismatic proponents will argue that expressions of legitimate piety may be introduced into the Mass, according to the Church’s canonical tradition (see Code of Canon Law, can. 22ff.). And they will note, for example, that praying with hands upraised has taken place in many Masses in Vatican City and elsewhere in Rome at conferences recognized and overseen by the Holy See, and that this pietistic practice is also biblically well-grounded (see Psalms 28:2, 63:4, 134:2, 141:2).
They will make similar arguments for praying in tongues, although given how strange this practice is to most Catholics, its expression should be limited to Masses specifically offered for Catholic Charismatic groups. (One might make the same argument, though less strongly, regarding praying with upraised arms.) Whereas in past years Catholic Charismatics would pray in tongues during the consecration at Mass, there’s been a trend to refrain from doing so and praying in this manner only during the Gloria.
Some debate the authenticity of praying in tongues, referring to it as “pious gibberish” and distinguishing it from the gift of speaking in tongues. Others will cite St. Paul to support the gift’s authenticity, which involves praying in unintelligible utterances.
Unlike the other charismatic gifts, praying in tongues is difficult to authenticate in practice. Praying in tongues is also described by charismatic leaders as a gift that most people open to its reception receive. In any event, no one should feel compelled to seek or display a particular gift or prayer style.
Finally, a distinctive action that all Catholic Charismatics experience is being prayed over to receive the “baptism of the Holy Spirit.” This term can cause confusion, because it is not a sacrament, and one undoubtedly receives the Holy Spirit at baptism. But Catholic Charismatics describe the experience as stirring up the gifts one has received at baptism and confirmation and thus an increased experience of the Spirit in their lives and thus a fuller activation of the aforementioned gifts.