- Cases of confirmed fraud (prosecuted or not): 10 (0.4%)
- Cases of confirmed fraud or non-citizen voting (which would be caught by checking drivers licenses): 52 (2.5%)
- Total confirmed illegal votes by non-felons (which puts an upper bound on our data): 239 (11.6%)
~~~~ How many known cases? ~~~~
This article appears to be the most in-depth investigation of the topic.
A News21 analysis of 2,068 alleged election-fraud cases since 2000 [until 2010] shows that ... in-person voter impersonation on Election Day, which prompted 37 state legislatures to enact or consider tough voter ID laws, is virtually non-existent.
In an exhaustive public records search, News21 reporters sent thousands of requests to elections officers in all 50 states, asking for every case of fraudulent activity including registration fraud, absentee ballot fraud, vote buying, false election counts, campaign fraud, casting an ineligible vote, voting twice, voter impersonation fraud and intimidation.
Their conclusion was that very few cases of fraud were of the type that could be prevented by voter ID laws. Most cases involved absentee ballot or registration fraud (where your ID would not be checked). Those that involved in-person fraud were usually of the "able to vote twice" or "am I eligible in the first place" variety, which is also not resolved by proving that you are who you're supposed to be.
Their database only has 10 cases of in-person voter impersonation fraud, which is the only kind that a loose voter ID law would catch (i.e. any proof of ID). A stricter form which required a type of ID which only proven citizens could get (such as a driver's license) would have eliminated 52 more cases of (probably) unintentional fraud. Taken together, these account for 2.5% of all reported cases over a ten year period.
There's also their 187 cases of generic "Casting Ineligible Vote" by Voters, which includes some felon voting (not caught by voter ID), some non-citizen voting (which is included above), but mostly there isn't information on what type it was. If you assume this extra set is entirely the kind of illegal voting that would be caught by voter ID, and that there's no overlap with the above set, that raises the total to a maximum of 239 cases of illegal voting (11.6%).
~~~~ So what is the impact? ~~~~
Under this worst-case scenario, where all of the 239 cases reported come from separate instances of detectable voter fraud, it's still not enough to have any significant impact on the majority of elections.
These cases were spread throughout much of the country, over a twelve year spread. But even if they were all concentrated in one district, in one year, they still wouldn't be enough to have an impact on any national-scale election. For example, the smallest districts (on average) have around 500,000 people. Even assuming that only 125,000 (1/4th) of those are actually registered voters, and there's only 10% turnout, that's still 12,500 votes cast. A 51-49% split would have the winner win by 250 votes, which is more than the 239 known cases of fraud.
To put it another way, there were 351,971,792 votes cast (total) in the presidential elections in the years in question (2000, 2004, 2008) (2012 was too late to be included in the database). 239 votes represents 0.0001% of that total.
~~~~ References & Notes ~~~~
I will point out that I only looked at national-scale elections: Representatives, Senators, and the President. It's possible that local elections (for mayor or other city/county positions) were decided by a small enough margin that a few people voting fraudulently (deliberately or not) would have swung the result. However, I'm not going to look into the data that closely, and the smaller the vote, the harder it will be to vote too often.
Finally, I'd like to also reference this answer, which points out that in order to have even 1000 deliberately fraudulent votes:
[E]ither one person has to travel to and vote at a 1000 booths to supply a 1000 votes, or a 1000 people have to collude to vote above and beyond any normal legal incentives to vote. Which is why Tammany Hall corruption was quite visible.
Note: The database is as comprehensive as possible, but is not 100% complete. See here for details, but basically not all government officials responded. However, if the government didn't respond, any fraud they may have had cannot be considered "confirmed".
You can query their data directly here, as well as images of every document they sent or received as they were building it.