I'm not able to delete -> in {{x -> a}, {x -> b}, {x -> c, x -> d}}

DeleteCases does not by default operate on heads. You can set the Heads option to True to change that.

DeleteCases[{{x -> a}, {x -> b}, {x -> c, x -> d}}, Rule, {3}, Heads -> True]

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

Note that the last term has become {x, c, x, d} which is not quite what you expected, but it is logically consistent if we expect {x -> a} to become {x, a}.

A simpler path to the same output in this case is:

{{x -> a}, {x -> b}, {x -> c, x -> d}} /. Rule -> Sequence

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

Your expected output can be had from:

{{x -> a}, {x -> b}, {x -> c, x -> d}} /. Rule -> List /. {x_List} :> x

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

Recommended reading:

• How to completely delete the head of a function expression

Replacing heads is generically supported by Apply (@@ / @@@). So a functional way may look like this:

rules = {{x -> a}, {x -> b}, {x -> c, x -> d}};
If[Length[#] == 1, Flatten[#], Identity[#]] & /@ Apply[List, rules, {-2}]

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

Or

If[Length[#] == 1, Sequence @@@ #, List @@@ #] & /@ rules

returns the same result.