Lava tubes are good candidates for habitats on the Moon because they are ready made structures with dozens of meters of solid basalt shielding. In addition to radiation shielding, these environments are also thermally stable (estimated temperature -20° Celsius / -4° Fahrenheit). Lightweight, compact inflatables can be used at first to provide habitable space. Eventually, It should be possible to seal cracks in the lava tube walls to provide an airtight, shirt-sleeve environment. Some lunar lava tubes may be large enough to contain small cities. Living in a lava tube provides many advantages.
Lava tube advantages
- There is substantial shielding due to having several meters of solid basalt between you and the outside. This is good against micrometeorites and radiation.
- There is heat insulation because you are underground.
Lava tube disadvantages
For her doctoral thesis under advisor B. Ray Hawke at the University of Hawaii, Cassandra Coombs scoured Lunar Orbiter photographs to identify a number of likely sites for lunar lava tubes. Due to the limiting resolution of the images, only the largest candidates could be identified. It is likely there are many smaller lava tubes in the identified areas and elsewhere. Smaller, but still large by human scale.
A 1969 paper by Oberbeck, Quaide, and Greeley, taking into account only the Moon's lighter gravity, and not the stronger nature of lunar anhydrous glass, calculated lunar lava tubes could reach 340 meters in span (lengths could go to kilometers). There are some indications in Coombs and Hawke's work (see "Locations" below) that some lava tubes may be much larger.
An analysis of lava tube shielding from Solar Proton Events, Galactic Cosmic Rays, etc., has been done by De Angelis et al. When it comes to shielding, a little may be worse than nothing, because primary particles can interact with shielding material to produce a spray of secondary particles. Additional shielding is necessary to protect against these daughter particles. Findings by De Angelis et al. are that one meter of shielding is sufficient against Solar Proton Events, but powerful Galactic Cosmic Rays need roughly six meters for effective shielding. Their analysis took into account that the upper five meters is regolith. The findings were, "The radiation safety of lunar lava tubes environments has been demonstrated." 
As early as 1957, the prescient Robert A. Heinlein described people living and working in giant volcanic voids in the Moon (although they were large bubbles rather than lava tubes). In the 1976 second edition of Depths of the Earth, speleologist William R. Halliday suggests large lava tube caves may exist on the Moon and Mars, and "mankind may look to these. . .caves for ultimate survival." NASA "experimental planetologist" Friedrich Hörz finally put it all together in his 1985 paper, "Lava Tubes: Potential Shelters for Habitats". Oregon L5 Society expanded on this work, holding analog exercises in lava tube caves with Young Astronauts and filling-in research topics such as "Lavatube Entrance Amelioration on the Moon and Mars" and "Moon Lighting: Illumination for Lunar Base Construction and Operations". Their multi-year effort to publicize and bring the lava tube option to the attention of NASA and space development enthusiasts in general has been effective.
Expanded Data (all the data we can find)
- Coombs, C.R., and B.R. Hawke, "A Search for Intact Lava Tubes on the Moon: Possible Lunar Base Habitats", in The Second Conference on Lunar Bases and Space Activities of the 21st Century, W.W. Mendell, ed., NASA CP-3166, v. 1, p. 219, 1992.
- Blacic, J.D., "Mechanical Properties of Lunar Materials Under Anhydrous, Hard Vacuum Conditions: Applications of Lunar Glass Structural Components", in Lunar Bases and Space Activities of the 21st Century, W.W. Mendell, ed., Houston, Lunar and Planetary Institute, p. 487, 1985.
- Oberbeck, V.R., W.L. Quaide and R.G. Greeley, "On the Origin of Lunar Sinuous Rilles", Modern Geology, v. 1, p. 75, 1969.
- Giovanni De Angelis, J.W. Wilson, M.S. Clowdsley, J.E. Nealy, D. Humes, and J. M. Clem: “Lunar Lava Tube Radiation Safety Analysis”. Journal of Radiation Research, Vol. 43, S41-S45 (2002) 
- Heinlein, Robert A., "The Menace from Earth", in collection The Menace From Earth, Signet, pp. 92-115, 1962 (orig. story copyright ©1957).
- Halliday, William R., M.D., Depths of the Earth, Revised and Enlarged Edition, Harper & Row, pp. 377-397, 1976.
Resources (not used but potentially relevant)