The Rules Expansions section of the site is newly revised following discussion on OD&D74.
Historic Rules 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 they will not necessarily match other versions of the game). Higher levels begin to require more work as magic-users would have the undescribed 3rd level spells, 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.
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.
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. It's unclear whether or not they simply added the word "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".
Thanks for Falconer on OD&D74 for inspiring the above list.
Modern Rule Expansions
Holmes Companion by Meepo - a concise 4-page expansion covering levels 4-9
Holmes Level 12 by aldarron and greyharp - thread describing creation
Holmes House Rules Supplement by oltekos
Holmes Treasury by delve
The Gray Book by Traveller
Holmes77 by RC Pinnell aka Thorkhammer (plus a rules expansion called Classic77)
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
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
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
Blood of Prokopius: Holmes Thought Experiment series (Holmes plus Cook/Marsh Expert)