Basic-Special combos are sequences of attacks composed of a light or heavy basic attack combo that allows you to follow up with a Special Attack that cannot be blocked, when most of the cases this will cause the special to miss. This does not include the passives of Green Arrow/Arrow or Martian Manhunter, and should not be confused with Unblockable attacks (those will break and negate blocks, while opponents are unable to block at all against these).
At one point, a hidden update made it impossible to use specials at all while your opponent is knocked down. After much player complaint, it appears that some of them have been reinstated.
The effectiveness this brings to Wonder Woman/Red Son and Superman/Red Son contributed to their popularity. Red Son Superman was the only character known who can make both of his specials unblockable this way, until the 2.6 update allowed Godfall Superman to do the same. They were later joined by Batman/Dawn of Justice.
Usually, specials that can be used this way have fairly fast starting animations and the character's heavy combo does not knock down/back the opponent too far.
Currently, it seems only combos that involve heavy basic attacks can be used. Specials cannot be used at all while your opponent is knocked down by light basic attacks.
Special 1: Wonder Woman (all versions except Dawn of Justice), Superman (all versions), Batman/Arkham Knight, Batman/Dawn of Justice, Batgirl/Cassandra Cain, Harley Quinn/Suicide Squad, and Doomsday (all versions).
Special 2: Superman/Red Son, Superman/Godfall, Shazam, Static, Batman/Arkham Origins, Batman/Red Son, Catwoman/Batman Returns, Batman/Dawn of Justice, Doomsday/Blackest Night, and Green Lantern/Jessica Cruz Rebirth.
Previously, it is possible to chain Superman/Red Son's special 2, Static's special 1, Doomsday/Prime's special 2, Sinestro's special 1 and Batgirl/Cassandra Cain's special 1 into their light basic attack combos (Batgirl, Doomsday, and Superman can still chain their SP1s into their heavy attacks).