What are the rules for using the labels Jr., Sr., III, etc. in a name?
Three traditional rules govern these labels (although often ignored in modern usage):
1. A son drops the 
Jr. label soon after his father dies — as Justice Oliver Wendell Holmes did. But there are two exceptions to this rule. First, if the father was especially famous, the son may retain 
Jr. {Frank Sinatra Jr.} {Hank Williams Jr.}. (Holmes was unmoved by this rationale.) Second, a son who earns an honorary title, such as 
Dr., 
Admiral, etc., should drop 
Jr. even if his father is still alive because the title differentiates the two men, making 
Jr. unnecessary.
The dropping of 
Jr. has long been considered proper form. But today many an octogenarian keeps the 
Jr., perhaps from a misplaced filial piety and the hope of preserving memories. Let no one doubt, however, what all the etiquette mavens have long said:
	- 1937: “At the death of his father, he is no longer junior.” Margery Wilson, Pocket Book of Etiquette 105 (1937).
	- 1938: “After the death of the father, the son drops the ‘junior’ from his name.” Mrs. Cornelius Beeckman, Common Sense in Etiquette 72 (1938).
	- 1955: “A man is ‘Mark Strand Jr.’ only while his father is alive and, of course, he bears his father’s exact name.” Nancy Loughridge, Dictionary of Etiquette 105 (1955).
	- 1963: “[A] man is ‘junior’ only while his father is alive and, of course, only if he bears the identical name.” Amy Vanderbilt, New Complete Book of Etiquette 594 (1963).
	- 1990: “Names are traditionally numbered only among the living.” Judith Martin, Miss Manners’ Guide for the Turn-of-the-Millenium 60 (1990).
	- 2003: “When the father dies, the son usually drops the ‘Jr.’ within two years’ time, since there is no longer a confusion between the two men — unless, of course, the man is the son of a very famous man . . . .” Letitia Baldridge, Letitia Baldridge’s New Manners for New Times 653 (2003).
2. The names must be identical to use 
Jr. and 
Sr. So the second Bush president (George W. Bush) is not a junior because his father’s name is different (George Herbert Walker Bush). Yet for journalists and the public it has become common shorthand to use “George Bush Jr.” and “George Bush Sr.” for easy identification.
3. When a man shares a name with both his living father (Jr.) and his living grandfather (Sr.), he may adopt the label 
3rd or 
III. If the grandfather dies first, the grandson becomes 
Jr. If the father dies before the grandfather, the grandson stays 
3rd or 
III. After both the father and grandfather have died, the grandson drops all labels.
Yet the issue that most often arises is punctuation: comma or no comma before 
Jr.? Journalists prefer omitting the comma — and this approach is logical because the label actually is restrictive. That is, the 
Jr. is a differentiator, not an appositive. Men who bear these labels, however, seem to prefer the comma. Although both forms are acceptable, LawProse’s in-house style is to omit the comma {Jack M. White Jr.}. Not only does the comma-less 
Jr. make possessives easier {Jack M. White Jr.’s book}, but it also looks cleaner. Strunk & White preferred it that way. So should you.
Sources:
William Strunk Jr. & E.B. White, 
The Elements of Style 3 (3d ed. 1979).
The Chicago Manual of Style § 6.47 at 322-23 (16th ed. 2010).
Garner’s Modern American Usage 555-56 (3d ed. 2009).