What are the differences between 报名，登记，挂号？
Definitions:

报名
To fill in personal information to the organizer in order to participate in the activities.
（参加活动时，向主办单位填写个人资料。）

登记
To record the relevant matters in the form or the booklet of the organizer or the competent authority; to register for a service (such as to subscribe to a delivery service or a social networking website).
（将相关事项记录在主办单位或主管机关的表格或册子上。）

挂号
To get a number in order to do something in sequence.
（取得编号，以便依序进行某事。）

Differences:

挂号 must involve the number, however 报名 and 登记 do not have to.
（“挂号”必定牵涉到号码，而“报名”和“登记”则不一定。）

报名 usually 登记 some information, and 登记 may not be a 报名, so 登记 is broader than 报名.
（“报名”通常会“登记”一些资料，而“登记”不一定是“报名”，所以“登记”的范围比较广。）

The relationship between 挂号 and 登记 is relatively close, because 挂号 usually 登记 some information in order to distinguish.
（“挂号”和“登记”的关系比较密切，因为“挂号”通常要“登记”一些资料，才能区别。）

报名 and 挂号 are less relevant. We use them in different areas, and do not mix them up.
（“报名”和“挂号”的关系比较小。各自用在不同的事件。我们不太会把这两个词混用。）

Examples:

报名
考试 examination or test
比赛 competition or game
培训 training or course
加入团队 to join a team

挂号
看病 to see a doctor
寄信 to send a mail
接受服务 to receive service

登记
出生死亡 birth or death
收养 adoption
结婚离婚 marriage or divorce
不动产 real estate
土地房产 land or property
车辆 vehicle
开业 to open a business