The major part, and the majority ought not to be used when a plain most would meet the case. They should be reserved for occasions when the difference between a majority and a minority is significant. Thus:
Most of the members have been slack in their attendance.
The majority of members are likely to be against the proposal.
When the word is used in an abstract sense it should have a singular verb ("a two-thirds majority is necessary"); where it denotes the people composing the majority the verb should be plural ("a majority of the committee were against the proposal").