Political support for reforming gun laws
It is unclear to me that the average citizen is unconvinced to leave gun laws largely alone. Note that the average citizen is not voting for candidates who favor more restrictive gun laws. Republicans currently control all three elected portions of the government (and the unelected Supreme Court). Yes, Hillary Clinton won a plurality of the popular vote, but Donald Trump plus Gary Johnson won more votes than Clinton plus Jill Stein.
Another argument is that polling shows that citizens support various reform proposals when asked. But this support tends to be soft. Very few people vote based on that.
All gun sales should go through a background check.
Most people support this. The Manchin-Toomey compromise was based on this. It was rejected by the National Rifle Association. The public reason was that it had insufficient protections against a national gun registry. Most NRA-friendly politicians voted against it, so it did not pass.
The average citizen is for this in principle, but I haven't seen a big hew and cry over it. It has not passed and would not affect most of the newsworthy events. If it were to pass, it would have minimal effect on gun rights. It is very seldom that someone legally obtains a gun without being able to pass a background check.
Ban assault weapons.
This is a bit tricky. One simple argument is to simply point out that anything that actually is an "assault weapon" is already under severe restrictions. Note that actual assault rifles are almost never used in crimes. Instead, people tend to buy lookalike guns like the AR-15.
Another argument is that we tried that. It didn't work. There are other guns that perform the same that are still available, as the differences between an "assault weapon" and any other rifle are mostly cosmetic:
The latest version of Feinstein's bill covers any semiautomatic rifle with a detachable magazine if it also has a pistol grip or forward grip, a grenade launcher or rocket launcher, a barrel shroud, a threaded barrel, or a folding, telescoping, or detachable stock.
No mass shooting has used a grenade launcher or a rocket launcher, as the ammunition is not generally available. Banning those is a solution in search of a problem.
Semiautomatic is the most common type of gun in civilian use. (Individuals in the military use fully automatic weapons. Plus there are larger weapons that don't fit well in the categories used by weapons that can be operated by a single individual.)
For the rest of it, all those things are within the capability of a 3D printer these days. Yes, actual 3D-printed guns don't work that well. But those aren't 3D-printed guns. They're parts frequently made of plastic or wood. They are exactly the kind of things that can be 3D-printed. The hard part to 3D-print is the receiver. The receiver is not restricted by the ban, so people can just buy any workable receiver and build an "assault weapon" around it.
As I said earlier, very few people obtain weapons legally that cannot pass a background check. So an expanded background check system would have minimal effect. The real problem is that crazy people are not getting put into the background check system as too crazy to have a gun.
We used to have an assault weapons ban. There is no evidence that it stopped gun violence. Most gun violence is with handguns, not any type of rifle. More than 60% of all firearm homicides and 90% of homicides where the type of firearm was identified are committed with handguns.
One thing to watch for in anti-gun arguments is misleading switches. They will talk about gun deaths, gun homicides, and total suicides all together, switching as necessary to grab the right statistic. Consider:
Gun deaths are highest in states with the most guns.
This is true, but only because gun suicides are higher in states with the most guns. Two out of three gun deaths are by suicide. Only one out of three are homicide. Only about 4% are something else.
They will often state this in such a way that someone might think that gun homicides are highest in states with the most guns. However, that's not true.
Reducing the number of guns reduces gun suicides.
This is certainly true. However, who cares? If we're talking about reducing suicides, we should be talking about all suicides. And there's no reason to think that reducing the number of guns will have a long term effect on suicides.
In Australia, they confiscated all the semiautomatic guns. This reduced the number of gun suicides the next year by more than would have been expected (Australia was in the middle of a ten year drop in gun suicides). Two problems though. First, the number of suicides by hanging was unexpectedly high that year. Two, the number of gun suicides jumped the year after that, once people had time to replace their semiautomatics with revolvers and shotguns. Overall, in the five years after the ban, gun suicides dropped by about as much as they had in the five years prior to the ban.
Often cited as a great success for gun control, it as actually something of a fizzle. Negligible effect on gun homicides, although neither Australia nor New Zealand have experience a mass shooting since then. It's unclear why confiscating guns in Australia would have affected New Zealand. Presumably they have something else in common. Or random chance.
In Japan, there are no gun suicides. However, total suicides are higher per capita than in the United States.
If there is one country that has been really effective in preventing gun violence, it is Japan. They have a complete ban on the sale of guns and ammunition. Even organized crime doesn't try to use guns in Japan.
Consider how they did it though. Obviously they don't have a second amendment. But that's not the real advantage that Japan has. They don't have an exclusionary rule (which derives from the fourth amendment). So if a police officer sees a person with a gun run into a group of people, the officer can require the entire group to submit to searches.
In the United States, a police officer could only search the single person who had been carrying the gun. If a police officer searched everyone, the possession charge would be thrown out as the search would be excluded as having insufficient basis. The officer could only search with probable cause, and "one of them must have it" is not considered probably cause.
In Japan, if the search finds its target, then it was clearly justified. The result takes precedence.
Even with the second amendment and widespread gun ownership, elimination of the exclusionary rule in the United States would allow police a much greater ability to control crime. Some libertarians may find this too risky (libertarians generally favor the exclusionary rule and the fourth amendment), but this argument puts the onus on leftists to explain their support for a rule that contributes to gun violence.
Another argument is that the second amendment prevents many types of effective regulation. In particular, regulations can't significantly reduce the number of guns in circulation. All they can do is slow the rate at which guns are added to circulation. This is because when a gun is banned currently, they can't take it away from existing owners. So everyone that currently has the gun keeps it. People just can't buy new ones in stores.
The argument here is that they should really focus on effective changes. However, effective changes require repealing the second amendment. So they should focus on that.