Despite a begrudging respect, there are few heroes with less in common than Wolverine would make short work of Peter Parker, a review of their comic histories actually suggests the Wall-Crawler would find it easier to dispatch the former Weapon X than vice versa.
But how is that possible? While Spider-Man has enhanced healing compared with the average person, Wolverine's healing factor is the stuff of legend, capable of reassembling him in a matter of minutes even when his skin and even musculature have been burned clear away (as evidenced in Wolverine #44 by Marc Guggenheim and Humberto Ramos, when Logan took on exploding super-villain Nitro.) But as many Marvel stories have shown, Wolverine can die, and Spider-Man's unique abilities actually make him a major contender for taking down the brutal mutant hero.
Repeated brain injury
This first entry on the list may not be a straight kill, but it does highlight Spider-Man's main advantage: his spider-sense. Peter Parker's spider-sense allows him to sense - and automatically react - to incoming physical danger. Wolverine has taken a beating from several telepaths - most notably recurring villain Mister X - because they're able to react to his moves before he makes them. The only reliable way around this has proven to be entering a mindless berserker rage, but that wouldn't work on Spidey, whose powers aren't based on mind-reading.
Wolverine would struggle to take down Spider-Man, and that means Spidey has time to work his magic. While Logan can heal from many injuries, the process isn't instant, and he often loses consciousness once he's taken enough damage. In World War Hulk: X-Men, the Hulk's Green Scar persona speculates that enough repeat blows to Wolverine's head will cause irreparable damage as his brain is slammed against his Adamantium skull.
Spider-Man actually has experience with doing this kind of damage, with his longtime villain Doctor Octopus developing a fatal condition in Amazing Spider-Man #600 after years of traumatic brain injury at his fists. Whether or not it would be possible to permanently incapacitate Wolverine this way as Hulk suggests or whether it would just take a lot of healing, Spider-Man is definitely capable of doing enough damage that Wolverine would be out for the count, at which point everything else on this list becomes even easier to pull off.
Suffocation and/or drowning
Wolverine's most definitive comics deaths have come due to loss of oxygen to the brain. Death of Wolverine ended with Wolverine being coated in adamantium and suffocating to death, and Wolverine also killed his son Daken, who shares his healing powers, in Rick Remender and Phil Noto's Uncanny X-Force #34 by drowning him in a shallow puddle.
Spider-Man is already significantly stronger than Wolverine, meaning that if there are any bodies of water around the pair when they fight, Logan's already a goner, but even if not, Spider-Man also carries around a rapid-setting adhesive that could cut off Wolverine's oxygen if Spidey was prepared to get in close. Spider-Man's web fluid is incredibly strong, but it can be cut, so Pete would have to exhaust or otherwise incapacitate Wolverine beforehand for this to be fatal.
Remove his powers
Wolverine's healing powers don't include magical resurrection, just rapid healing, which means that if Wolverine is slain while his powers are inactive, that death will stick. Marvel Comics have shown many different ways of negating Wolverine's powers, but the most relevant to Spidey are the mutant inhibitor collar and Carbonadium (a radioactive metal that suppresses Logan's healing.)
Given his s in the spy community - including his secret agent sister Teresa Parker - it wouldn't be hard for Spider-Man to acquire an inhibitor collar (used frequently for mutant restraint, and shown to negate Wolverine's healing in the Civil War tie-in story "Vendetta") and his science skills would likely allow him to synthesize his own Carbonadium, though prior adventures suggest he could also locate and steal either if needed.
Incineration and irradiation
While Wolverine has survived several major bombings - a purported nuke in 2004's Venom #8 and an atom bomb (with some protection) in 2008's Logan #1 - he has also been killed via incineration, as famously depicted in Chris Claremont and John Byrne's "Days of Future Past" story. This is a contentious topic for fans after Logan was shown to survive the previously mentioned attack by Nitro, but the deciding factor appears to be his bone marrow.
Given enough energy to draw on, Wolverine has been able to regenerate from a drop of blood - taking place in the decidedly unusual situation presented in Claremont and Alan Davis' 1987 Uncanny X-Men Annual - so for incineration to work, destroying or breaking down Wolverine's bone marrow is key (something irradiation could also accomplish.) Again, Peter's scientist side is more than capable of inventing a gadget to do the job; the man who discovered Alpha Energy and invented the Spider Armor Mk. III is more than capable of constructing armaments on the level of the Sentinel blast that incinerated Logan in Uncanny X-Men #141, or just converting his own radioactive blood for offensive use.
Interdimensional marooning
As a member of the Web Warriors - the spider-themed heroes who guard the "Spider-Verse" from chaos - Spider-Man has on multiple occasions been given possession of a dimensional travel watch (as well as the chance to study how they work.) Granting access to the web of life, the watch would allow Spider-Man to drop Wolverine in a dimension of his choosing - whether that be an all-water world, a dimension as hot as the sun, or one of the many zombie-outbreak universes with a strain capable of taking Wolverine down (the one in Marvel Zombies did the trick.)
Spidey is actually one of the Marvel Universe's greatest minds - not the equal of Reed Richards, Marvel's smartest man in the world, but capable of contributing to his research on countless occasions - and so the possibilities of time travel and teleportation also exist. Al Ewing and Joe Bennett's The Immortal Hulk has confirmed that translocation requires advanced mathematical calculations of Peter's intellectual level to perform properly, and his colleague Grady Scraps developed a time travel device while working with Pete at Horizon Labs (in "Trouble on the Horizon" from Dan Slott's run on Amazing Spider-Man,) meaning that Spider-Man's ability to teleport Wolverine deep underwater or go back in time and kill him before his mutant powers activate are more a challenge of resources than capability.
Feed him to the Inheritors
During his time with the Web Warriors, Peter's main challenge was the Inheritors - Hulk-strength predators who survived by consuming the life energy of Totems - beings who symbolically connect the human and animal kingdoms. While Spider-Man is an impure Totem, his gifts given to him by radiation, the Inheritor Morlun was able to feed on him in J. Michael Straczynski and John Romita Jr.'s Amazing Spider-Man #35.
It stands to reason that Wolverine - even more connected to his animalistic totem than Spider-Man himself - would be prime food for the Inheritors. Given that the Inheritors self-resurrect using cloning technology, and the Web Warriors have had to actively work to stop them returning in full force after their defeat in Christos Gage and Jorge Molina's Spider-Geddon, this technique might actually be the easiest to pull off, as Spidey would simply have to make the Inheritors aware of Logan and then stay out of their way.
The Enigma Force
Though the least likely entry on this list, Spider-Man has drawn on the powers of the Enigma Force to become Captain Universe in the past - originally in Gerry Conway, Sal Buscema, and Mike Esposito's Peter Parker, The Spectacular Spider-Man #158. Granting him the power to punch the Hulk into orbit, the Enigma Force would easily allow Spidey to overpower Wolverine, as well as to enact any of the techniques mentioned above.
Spider-Man doesn't have unfettered access to the Enigma Force - or any truly reliable way to summon it - but he does have a connection to the Force that has seen it enter his life again and again, meaning that in the right set of circumstances, it could conceivably be an option for taking down the best there is at what he does.
(Possibly) Overload his healing factor
As any longtime Marvel fan knows, Wolverine has benefited from a degree of "power creep" over his comic history. While some stories suggest that sufficient damage will simply overtax Wolverine's healing factor, leaving him vulnerable, writers have generally moved away from this idea as his stories have continued.
Still, there have been hints that severe enough damage - cutting off Logan's head or directly piercing his heart - is enough to kill him off. Wolverine vol. 3 #20 (from Mark Millar and John Romita Jr.) and Age of Ultron #10 (from Brian Michael Bendis and various artists) certainly seem to suggest that Wolverine can be permanently killed by piercing his heart - perhaps an extension of his need for oxygen - though it's difficult to reconcile this with his ability to regenerate from blood and bone marrow.
Regardless, as already mentioned, Spider-Man is uniquely equipped to survive extended hand-to-hand combat with Wolverine, so if Wolverine's healing factor has an upper limit (or depends on caloric consumption to function) Peter would likely be capable of running out the clock, especially if he used some of his more advanced armor (e.g. the Iron Spider suit), unusual powers (e.g. the retractable stingers shown in "The Other" or his grotesque Mark of Kaine technique) or chose to use his super-strength to just hurl cars at his opponent.
Spider-Man is one of Marvel's sweetest, most empathetic heroes, so there's little chance that he and Wolverine will ever come to truly fatal blows outside alternate universes and eventually-erased futures. Still, it's fascinating to realize that while Logan might be the one with the killer instinct, Peter's long history and vast scientific knowledge actually make him a huge threat to a hero many fans consider unkillable.
