If you want to bring songs into your class but are not sure what to do with them, the first thing you can do is all the boring stuff that students usually hate. If you just add the music element they will happily indulge in exercises that usually make them groan and learn language points that usually scare them off. This is also great practice for exams, for example the FCE Use of English paper. Examples of 'doing boring things with songs' are:

Gap fills (Open cloze)
Match the sentence halves
Error correction
Put it into order- Words and lines
Put words into the correct form
Dictation
Pronunciation work

Gap fills (Open cloze)

Remove single words from the text, by tippexing them out on the page or replacing them with gaps in a Word document. Students try to guess what the missing words are and then listen to check. DO NOT remove random words and ask students to listen to fill the gaps without having even read the lyrics through first. As popular as this 'random gaps with random songs' task is with students and some teachers, it has no actual teaching aim. To make sure your activity does have an aim, make sure that:

you treat the text as a text, and have some kind of gist question before you move on to students filling the gaps, e.g. 'What is the relationship between the person who is singing and the person they are singing about?'
the words you remove from the text are guessable, meaning there are only 1 or 2 possible words that can go in each gap. This generally means removing 'grammar words' such as articles, auxiliary verbs and prepositions, but you can also test fixed expressions such as proverbs for higher level groups (e.g. every cloud has a ______ lining) and collocations such as 'make' or 'do' at any level.
the song matches your general syllabus in terms of topic ('She's leaving home' by the Beatles when you are doing the topic 'family') or grammar point ('I still haven't found what I'm looking for' by U2 for Present Perfect).
you have some kind of post-text speaking task such as discussion questions- about the meaning of the song, the students' opinion of the song musically or the students' opinions on the issues raised in the song (e.g. their opinions on drugs for 'Sorted Out for Es and Whiz' by Pulp).

To make the cloze task easier for students, you can list the missing words at the bottom of the page and just get them to put them back in the correct place. A (much more difficult) variation is to list the missing words but not show the gaps in the text where the words have come from ('I can see clearly now rain has gone'- the), which is good for auxiliary verbs (e.g. passives) and articles. The words can be listed at the end of the text or the end of each line.

Match the sentence halves

Split each line of the song into halves (e.g. If I had a hammer/ I'd hammer in the morning) and mix up the second halves of the lines. Students then try to put the lines back together, using grammatical and vocabulary clues, before listening to check. This is great practice for linking expressions and exam tasks where you need to put missing information back into a reading text. Again, make sure you have a lead-in and a post-text speaking task.

Error correction

Add errors into the text ('Pleased to meet you, hope you to guessed my name') and students try to correct it, then listen and check. DON'T randomly change things into other correct possibilities (e.g. When I'm 74), another totally pointless but popular classroom activity. At much higher levels (e.g. Proficiency), you can get students to correct the actual grammar the songwriter has used ('I ain't got nothing') - leading onto a discussion of the differences between informal and regional English and actual grammatical mistakes.

Put it into order- Words and lines

As long as you make sure students have some context in which to do the task (e.g. predicting the rest of the song from the first verse), you can ask them to either put the rest of the lines of each verse into order ('I really shouldn't care/ Who's in the flurry of her first affair/ Lord knows I'm not a fool, girl/ Lord knows I'm not a schoolgirl') or put the words of each line into order ('the can now I rain has gone see clearly'). The first is good for exam reading tasks, and the second is good for adjective and adverb word order.

Put words into the correct form

This is usually the most boring grammar task of all- put the words in brackets after each gap into the correct form. Again, add the song element and spend some time on what the song means and they will hardly notice the grammar. This is good for tenses (e.g. 'Rudolph the Red Nosed Reindeer _______ (have) a very shiny nose'), verb patterns ('Kept ______ (think) I could never ______ (live) without you by my side') and parts of speech ('Bye bye Miss _______ (America) Pie').

Dictation

Deeply unfashionable though it is, dictation really can be good occasional practice of listening and spelling. You need to set it up very carefully, but it can be done with songs, as long as they are very slow indeed (e.g. Mad About the Boy) or very repetitive. Alternative dictation techniques can also be done with songs, for example grammar dictation (also known as Dictogloss) or the activity where alternate students take down different word classes (e.g. one student listens for all the nouns and the other for all the verbs) then try to reconstruct the lyrics together.

Pronunciation work

If you put the gaps at the end of each line, students can use the fact that the words rhyme to put the correct words in the correct places. You can either list all the words at the end of each verse, or (more difficult) leave the end of each second line blank and let them think of possible rhyming words.

Once students are used to doing activities like those above in class, you can involve the class more by asking them to bring suitable songs in for a particular task, or even ask them to prepare tasks themselves to test the rest of the class with.

Specific songs for specific points- some suggestions

Sailing- Rod Stewart- Present Continuous
Nothing Compares 2 U- Sinead O'Connor- Present Perfect
I still haven't found what I'm looking for- U2- Present Perfect
Hand in My Pocket- Alanis Morissette- Adjectives (esp. of personality and feelings)
The Logical Song- Supertramp- word formation
I will survive- Gloria Gaynor- future
When I'm 64- The Beatles- future

Making the boring less boring

As I said above, the most important parts of a lesson with a song in are the lead-in and the post-listening task.

Lead-in ideas:

Listen to the intro to the song (the part before the lyrics start) and discuss what the mood of the song will be- Will it be about love or politics? / Will it be happy or sad? Students read the lyrics to check, then move onto the language task from the list above.
Do the same as above, but predicting things about the song from the title
The first time students listen, they only listen for certain words- for example, they put the 10 adjectives you give them in order as they hear them or count the number of times they hear a certain word or expression (e.g. 'I'm' for Hand in my Pocket)

Post-listening task ideas

Students discuss what will happen to the people in the song in the future
In pairs, students roleplay a conversation where the people in the song meet again, e.g. he asks her why she left and she explains. Discuss as a class what the two people said/ decided in each group.
Students write a final verse to the song

Naturally less boring

There are only so many gap fills you can do before you really ought to reward the students with something different- if only so they can't predict what the class will be like once you tell them they are doing a song. More 'traditional' things to do with songs include:

Sing along

Variations on this are:

Split the class into groups and have them sing alternate lines (obviously good for duets).
One team continues singing when the volume of the tape is lowered and gets points for how close they are to the timing of the tape when the volume is put back up again. This is good practice for sentence stress and rhythm.
Students sing quietly or loudly as the teacher holds up flashcards with 'up' and 'down' written on them.
Only students who match the sentence that the teacher holds up sing (e.g. 'Only students with long hair'). Students can then prepare similar cards for the next class.
Students lip-synch silently to the song and the class votes on who most looked like they were actually singing it. This is good for awareness of mouth positioning to make particular sounds in English, and for students who are self-conscious about their voices.

Listen and act

e.g. Head Shoulders Knees and Toes for small kids, but it is also possible to do this with less obvious ones like 'A few of my favourite things' from Sound of Music, especially if students can watch the film and copy the real actions.

Don't Really Do Songs At All

Finally, you can totally ignore the lyrics of the songs and just use it as a piece of music. Techniques include:

Discuss how the music makes you feel/ what adjectives you would use to describe it
Describe the scene of a film this would accompany, and possibly watch the real scene to check (e.g. Perfect Day from Trainspotting)
Identify the type of music, maybe from written descriptions of each type (e.g. the differences between different types of dance music, jazz or world music)
Have it in the background to set a mood
Match the music to stills from TV adverts or the actual adverts with the sound turned down (to lead in to discussions on advertising)
Match the song to the CD cover art
Match the (folk) songs to the nationality (to practice country and language vocab)
Match the song to the year (to practice dates)
Musical chairs