Historic Rule Expansions
1. Extrapolate from the Holmes Basic rulebook. Internally consistent rules for 4th level characters, at the very least, can easily be generated by extending the tables in Holmes, although the details will not necessarily match other versions of the game. 5th level requires more work as magic-users would have the undescribed 3rd level spells (if following OD&D), though some of these can be matched to magic items with similar effects. Hints for higher levels are sprinkled throughout the rulebook and the module B2. Here is a great example on Dragonsfoot by order99 about he expanded Holmes Basic when he didn't have access to other rulebooks.
2. Use OD&D. Dr. Holmes edited the Basic rulebook from the original D&D books and supplements, and thus the rules are in many ways closest to OD&D + some of Greyhawk. At the time of release of the Holmes Basic set in the fall of 1977, no AD&D rulebooks were available. Furthermore, the Holmes Basic set was originally packaged with the Monster & Treasure Assortment Levels 1-3, a product designed for OD&D. The M&TA assortment was eventually replaced by the module B1, which contained some further references to rules from Greyhawk. Thus, the OD&D rules are perhaps the most natural to use if expanding Holmes Basic. See this page for a reference sheet with character info bridging Holmes with OD&D up to name level.
3. Use AD&D. The Holmes Basic rules direct the player to AD&D for expanded play, including higher levels, in nine different locations. In Dragon #35 (March 1980), Gygax told us that the references to AD&D were added to Dr. Holmes' original draft by TSR. Now that the Holmes Manuscript is available, we know that TSR generally just inserted "Advanced" where Holmes had written "Dungeons & Dragons". The rules themselves are closer to OD&D than AD&D. A few new M-U and Cleric Spells that later appeared in the Player's Handbook in 1978 were added to Holmes Basic.
4. Use the 1981 Expert Set rulebook (edited by Cook/Marsh). In Dragon #35, Gygax also revealed that "Design is now hard at work on the second boxed D&D game, the Expert Set. It will take players through at least 12th level of experience, tie in the best of the “Original” material, and actually add some new classes, spells, magic, monsters and so on." In context, this appeared to be an Expert Set that would complement the Holmes Basic Set. A late 1980 Gateway to Adventure catalog shows the Holmes Basic set side-by-side with an unpictured but soon-to-be-released Expert Set. However, when the Expert Set finally arrived in 1981 it was accompanied by an entirely new Basic Set that replaced the Holmes Basic set (although TSR continued to have the original in stock through the Mail Order Hobby Shop until at least 1986). To aid owners of the original Basic set who did not wish to buy a new set, the Cook/Marsh rulebook contained a section on page X4 titled "Using D&D Expert with an early edition of D&D Basic" which began "If your copy of the D&D Basic rules has a blue cover with a picture of a dragon on it, then this section is for you". This section then provides a summary of the "new material found in the 2nd edition of D&D Basic". See this page for a reference sheet with character info bridging Holmes with Cook/Marsh up to sixth level.
Thanks for Falconer on OD&D74 for inspiring the above list in this thread.
Modern Rule Expansions
Holmes Companion by Meepo - a concise 4-page expansion covering levels 4-9 [link updated July 2013]
Holmes Level 12 by aldarron and greyharp - thread describing creation
Holmes House Rules Supplement by oltekos [no longer available at link]
Holmes Treasury by delve
The Gray Book by Traveller
Holmes77 (levels 1-9) by RC Pinnell aka Thorkhammer [link updated July 2013]
The Holmes Supplement: Portown by the same
Swords & Wizardry is an OD&D retroclone. The Core Rules 4th printing (May 2011) includes an optional "Blue Book" order of events for combat.
BlueBook for Swords & Wizardry by foxroe - S&W forum thread about this
Holmes-like D&D for Labyrinth Lord by 3d6 [link updated August 2013].
Mazes & Perils (levels 1-12). Nominated for a 2013 ENnie for Best Free RPG.
Blueholme Prentice Rules (levels 1-3) by Vile. Compleat Rules in progress.
Levels Beyond Basic (LBB) by bobjester; links to his blog Prime Requisite. See the far right side bar for the links to the LBB docs.
Holmes Ref - Reference Sheets for Holmes Referees - includes some expansion material, with more planned.
Resources for expanding the rules
List of changes made to the Holmes Basic rulebook
List of References Not Described
References to Monsters, Weapons, Magical Items, Higher Level Play not described in the rulebook, and references to other rulebooks
Typical Monster Dexterity Scores
See also:
Jeff's Gameblog: Expanding Holmes Parts I & II (using the Blue Book alone)
Hill Cantons: Holmes Expert Boxed Set: What If? - More on the Holmes Expert Set
Semper Initiativus Unum: Blue Book D&D (Holmes plus Cook/Marsh Expert)