Updated EPSG information as of February 27, 2021. Check the edit history for the previous version. Updated by mkennedy.
The Wikipedia article is out of date. The current WGS84 geoid model is EGM2008 which comes in two public versions: 2.5' x 2.5' and 1' x 1'.
WGS84 itself has has several realizations such as G973 and the latest, G1762, which are linked to various International Terrestrial Reference Frames (ITRF). Confluence GIS has a nice table on them. With each new realization, the latitude, longitude and ellipsoidal height values are little more accurate, and more closely match the corresponding ITRF to the point where they're matching within centimeters if not better in some places.
The EGM geoid models contain the difference between ellipsoidal heights (usually 'h') and geoidal heights ('H'). The differences are also called the geoid undulations ('N'). Thus you can use the model to convert between the two heights. Geoidal height is close to an actual elevation, and reflects how fluids would respond to gravity. When using ellipsoidal heights, water can flow uphill.
The geoid models are not accurate, not precise enough for the differences between the different WGS84 realizations to matter. The best, public geoid model available still has grid spacing of 1 by 1 minute. At the equator, 1' is around 1.855 kilometers.
If you knew you had EGM84-derived geoidal heights, you might want to convert back to ellipsoidal heights using the EGM84 model, then use a EGM2008 model to convert back again, but not much data that would have used EGM84 is accurate enough to bother doing that.
So far EPSG (IOGP's geodesy subcommittee) has only had one version of WGS 84, although using 3 different coordinate systems (axes/units): 4326 (geographic 2D, lat-lon), 4979 (geographic 3D, lat-lon-h), and 4978 (geocentric, XYZ). The subcommittee has added all 3 versions (geocentric, geographic 2D, and geographic 3D) for all known realizations of WGS 84. The geocentric and geographic 3D versions were added around 2016 while the geographic 2D versions were added in 2019, after the ISO geodetic registry added them.
Both registries (EPSG incorporates aliases/duplicate entries to keep up-to-date with the ISO registry) have some time-based conversions that convert between two WGS 84 realizations and other CRS.