The following argument was adapted from Alexander Pruss' Principle of Sufficient Reason, and to my knowledge there hasn't been any rebuttals to it.
The argument follows the basic strategy of ontological arguments- start with a very weak claim (e.g. it's possible that God exists) and then demonstrate that this weak claim leads to some stronger claim (e.g. since it's possibly necessary that God exists, then it's necessary that God exists by axiom S5 of modal logic). The premises of the argument are very basic intuitions about the causal structure of the world and should be uncontroversial.
(1) If an event E is in fact caused by C in the actual world, then E would not have occurred were no cause of E to exist
(2) Every contingent event possibly has a cause
(3) If, in the actual world, proposition q holds if proposition p holds (i.e. if p then q), then in a world where p does not hold, if p were to hold then q might hold
(4) If the propositions "if p then q" and "if p then not q" both are true, then it is not possible that p.
Discussion of each individual premise
If p's holding causes q to hold and not hold, that means p's holding itself is problematic, i.e. p would be impossible. 4 is obviously true.
2 is the weak version of a causal principle. Clearly, it's very difficult to see how to argue against it. All it's saying is that it's possible that all contingent events have causes. Humean intuitions, if true, may lead us to believe that it is possible that some events are uncaused. But the fact that we can conceive of some events being uncaused only proves we can conceive of the events being caused as well. So premise 2 also seems obviously true.
1 too is a rather inoccuous observation. It says that if in the actual world an event is caused, then in the actual world it could not have happened without a (not the) cause. This is not Kripkean origin essentialism which claims if C causes E, then C is a necessary condition for E. The claim here is not a modal one, and in fact is a lot weaker- if something is in fact caused in a possible world, then it wouldn't have existed in that world without a cause.
That leaves premise 3. It may seem complex, but this too is based on very basic modal and counterfactual intuitions. Let's begin the discussion of this premise by noting that it is similar to an axiom in modal logic, the so-called Brouwer Axiom. This states if a proposition is actually true, the necessarily that proposition is possible. In modal logic semantics, "p is possible" means p is true in at least one possible world, and "p is actual" means p is true in the actual world. Since the actual world is a possible world, p's holding in the actual world by logical necessity entails that it holds in at least one possible world, and hence is possible. Premise 3 encapsulates the similar observation that, in a possible world other than the actual world, the events of the actual worlds would remain as possibilities. In other words, in a non-actual possible world w, events that happen in some other world (in this case the actual world) are, in fact, possible.
So let's assume in the actual world, both p and q hold, and if p then q. Let's further assume that in a possible but not actual world w, p does not hold. Based on the discussion in the previous paragraph, the events of the actual world are possible in this world. So in this world, it might be that were p to hold, q would hold. Here's an analogy. Let's say Jones' setting fire to the barn (p) led to his getting arrested (q). All premise 3 claims is that in a world where both Jones didn't set fire to the barn, were he to have done that, he might have gotten arrested. So while this premise seems complex and circuitous, it is in fact as obvious as they come.
So with all of these obvious premises substantiated- as if they needed to be substantiated- let's enter the deduction.
The strategy here is reductio ad absurdum. We begin by assuming that some contingent event has no cause. We then demonstrate that this assumption, together with premises 1 through 5, lead to a logical contradiction. Hence, we conclude, the starting assumption was wrong and there can be no contingent event without a cause.
q is the true proposition that event E occurs, and
p is the true proposition that there is nothing that causes E (so in the actual world, if p then q)
Since premise 2 says every event can possibly have a cause, so the actually causeless E has a cause in a possible world. Meaning, p is false (doesn't hold) in some possible world w.
According to premise 3-
(5) In w where p does not hold, if p were to hold, then q might hold.
5 would become relevant near the end of the deduction again.
Since p does not hold in w, that means E has a cause in w. Call this cause C.
According to premise 1, in w, were no cause of E to have existed, then E would not occur. So in w, if p were to hold, q would not hold. In other words-
(6) In w, if p were to not hold, then that entails if p were to hold, q would not hold.
(7) In w where p does not hold, it is not the case that if p were to hold, then q might hold.
The move from 6 to 7 may seem abrupt, but saying "if p holds then q doesn't hold" is logically equivalent to saying "it is not the case that if p holds, q might hold".
But (5) and (7) are contradictory in just the way relevant to premise 4. Both 5 and 7 begins with "if p does not hold", but then
5 says- if p holds then q might hold.
7 says- it is not the case that if p holds then q might hold.
If you replace "if p holds then q might hold" with another proposition r, then 5 and 7 seem to be saying
(5) (If p were to not hold then) if p holds then r holds
(7) (If p were to not hold then) if p holds then r does not hold.
By premise 4, it is not possible that p holds. But p was assumed to be true, and true propositions are possible so our initial assumption says it is possible that p holds. This is a logical contradiction, so our initial assumption was false.
This completes the reductio, and since our initial assumption was false- it is not the case that a contingent event E can be causeless. In other words, every contingent event has a cause.
Here's an intuitive analogy to make the argument's force more tangible:
Let's say an airplane crashes due to metal fatigue in the ailerons. Now consider this counterfactual:
Were the plane hit by a surface-to-air missile, it would have crashed, and in that world were it not to have been hit, it would (or at least might) still have crashed (due to metal fatigue).
Now let's replace the "metal fatigue" explanation with "no cause" explanation. Let's say the airplane crashes due to no reason at all. So the equivalent counterfactual would be:
Were the plane hit by a surface-to-air missile, it would have crashed, and in that world were it not to have been hit, it would (or at least might) still have crashed (since it would have crashed for no reason at all).
But clearly this results in an absurdity. In the counterfactual world where the plane is hit by a surface-to-air missile, i.e. where the cause of the crash is the missile, it would still have crashed (for no reason). This contradicts the fact that in that world, the plane crash is caused by the missile. That means the original assumption- that the plane would crash for no reason whatsoever- is problematic, since it contradicts the proposition that in the world where the plane crashes the crash is caused by the missile.
It's a difficult analogy to get your head around, which is why the precise argument is required.
[By the way, I know the deduction is really clunky, I know, but that's probably because I chose not to use any notations. Here's the whole thing using notations.
Operators: (I'm making up some to adapt to blogspot's crappy formatting options)
(mgt)- if...then might
(evr)- for every x
(any)- for any x
(imp)- implies, close to entails
M- it is possible that
1. (C causes E)=>(~(evr)D (D causes E)->E did not occur)
2. (evr)E (E occurs (imp) M (any) C (C causes E))
3. (q&p& M~p) (imp) (~p->(p (mgt) q))
5. ((p->q) & (p->~q))=>~Mp
Let q be the true proposition that event E occurs
Let p be the true proposition that nothing causes E, that is ~(any)D (D causes E)
What is true is possible, hence
From 3 and 7-
8. ~p->(p (mgt) q)
Let ~p be true in w. In w, E has cause C. By 1, at w-
This is true in every world ~p holds. It follows from 9
p->~p is equivalent to ~(p (mgt) q). Thus by 4 and 10-
11. ~p->~(p (mgt) q)
By 3, 8, and 11-
6 and 11 are contradictory. Hence, the assumption that nothing causes E is false. This completes the reductio.]