Ukraine

Putin lacks internal support, runs a Mafia regime and is making unreasonable demands over Ukraine. NATO is a defensive organisation, not offensive. Therefore it's only a bulwark against possible aggression from Russia - the only credible enemy in the Europe and the reason NATO was set up in the first place.

The immediate admission of Ukraine to NATO - like tomorrow - would be a powerful statement and raise the stakes such that Putin's hand is weakened. No NATO troops would be required to be stationed in Ukraine (although Ukraine's troops would become de-facto NATO troops) - it's a deterrent that costs nothing beyond the normal cost of NATO.

Would Putin dare attack a NATO country? I very much doubt it - he wants Ukraine without a shot being fired and is hoping NATO and the EU will throw Ukraine under a bus. If he did attack, there would be overwhelming condemnation and a great risk of Russia being completely isolated and possibly being thrown out of the UN. The consequences are unthinkable for Putin and he won't want to be responsible for starting the next European war while risking internal revolution. 

If nothing else, it would buy more time. If Putin is appeased, there would follow a domino effect with the Baltics, Romania, etc, following very shortly, which is not in the best interests of anyone, except Russia. There is a deep moral argument for supporting Ukraine, but admitting it as a NATO member also solves the moral argument of sending other NATO troops there and putting them in danger.

Do nothing and the West is shown up as toothless and, by default, NATO too, despite Ukraine not currently being a NATO member. However, Putin could become sufficiently emboldened to next tackle the small NATO country of Latvia (population 2m), gauging that NATO would not risk war for such a small nation, which conclusion was reached not so long ago in a war games scenario. It's straight from Hitler's 1937-39 playbook.

Admitting Ukraine to NATO could work - there's to lose, except the possibility of showing NATO up as being toothless, in which case NATO is a useless deterrent and well past its sell-by date. Alliances have always been weak against a single enemy, as history has proven time after time. Only a brilliant alliance leader can prevail, usually by deceiving members of his own alliance (John Churchill being a prime example) into doing things they would normally balk at.

Even the demise of NATO could produce something far more potent - an EU Superstate with a combined military under a single command - exactly what Putin doesn't want; NATO is just an excuse - his real fear is Ukraine joining the EU and prospering. The effect of that on his own country would be destabilising to his own regime - which can only be a good thing for the Russian people.