Any built-in function to generate successive sublists from a list?

 lst={a,b,c,d};
 ReplaceList[lst,{x__, ___} :> {x}]

Speaking of "common operation":

 Table[lst[[;; i]], {i, Length@lst}]

A variant using Take.

list~Take~# & /@ Range@Length@list

{{a}, {a, b}, {a, b, c}, {a, b, c, d}}

One using NestList:

NestList[Most, list, Length@list - 1]

{{a, b, c, d}, {a, b, c}, {a, b}, {a}}

Subsets takes an optional 3rd argument as Subsets[list, {n}, k] that gives you the kth sublist of length n. Since your sublists are in sequence, you'll always need k = 1. You can then use this as:

MapIndexed[First@Subsets[list, #2, 1] &, list]
(* {{a}, {a, b}, {a, b, c}, {a, b, c, d}} *)

Another alternative would be:

Reverse@Most@NestWhileList[Most, list, # != {} &]