The UK government official parliamentary petition website allows citizens to submit and sign petitions.
Which petitions have ever succeeded? By succeeded, I mean been debated and then actually acted upon in favour of the wording of the petition.
There is an article that touches lightly on the topic here.
Using the content analysing tool BuzzSumo, I discovered the top ten most-shared campaigns from petition.parliament.uk within the last year. It seems logical that the most-shared petitions are those with the largest chance of success, and all reached the threshold to be debated by MPs (though four of the ten were actually denied a debate). Despite accumulating millions of signatures and hundreds of thousands of shares, however, not a single one of these campaigns suceeded in obtaining its intended outcome.
Beyond that you can see a full list of UK parliament petitions here. Any one that has over 100,000 signatures will likely be debated (some don't end up being debated, and some haven't been debated yet). If you click on a petition over 100k you should see a government response at the bottom that will directly address the yes/no outcome. I looked through all the ones over 100k (there aren't that many) and they all came up as either 'no', 'we're already doing this', or some sort of non-answer. Effectively, this means that no petition has managed to change parliament's mind.
The parliamentary committee which oversees the process issued a report in November covering the first year. They identified a few success cases:
Sugar tax - petition initially dismissed, but rolled into a larger ongoing debate around the topic; the government later announced such a tax would be brought in from 2018.
Meningitis B vaccination - after committee enquiry government agreed to provide a new public health campaign on meningitis symptoms, and to review how funding for vaccines is allocated.
Brain tumour research funding - petition prompted an enquiry, after which the government conceded the need for increased funding. This is now under review -
The Government agree that a greater level of brain tumour research is urgently needed [...] the Minister for Life Sciences will be convening a working group of clinicians, charities and officials to discuss how working together with our research funding partners, we can address the need for more brain tumour research.
So that's one success (where other campaigns, not just the petition, also contributed); one probable success but still being worked on; and one related positive change albeit not quite the thing asked for.
As none of these were apparently the "most shared" petitions in 2016 (for a start, one was in 2015), they won't have been picked up by the New Statesman article quoted in the other answer.
The review noted a fourth enquiry, the petition on high heels and sexism in workplace dress codes, but at the time of writing there was not an identifiable outcome from this. However, just last week the committee issued their report:
The government has said that the existing law is clear, and that the dress code that prompted this petition is already unlawful. Nevertheless, discriminatory dress codes remain widespread. It is therefore clear that the existing law is not yet fully effective in protecting employees from discrimination at work. We call on the government to review this area of the law.
So if this progresses further, there will be a fourth case to point to. Whether it will, good question, but IME it seems likely.
One thing to bear in mind here is the timescales. Two of the four examples here are on the way to a positive change that does what the petition wants, but are taking time to get there. And since we've only had a petition system since mid-2015, there won't be many done-and-dusted cases...
Although the current petitions area on the Parliament website dates from the last general election (mid-2015), there have been previous iterations of the same concept - i.e. officially sanctioned e-Petitions sites.
Prior to the current one, there was a very similar site in the 2010-2015 Parliament. I can't spot any petitions on that site that really led to any policy changing dramatically.
Going further back (before 2010), there were petitions to 10 Downing Street - i.e. to the executive rather than the legislature. In practice, most people signing such petitions probably don't think much about the difference.
I can't find any trace of the actual 10 Downing St petitions site, but from my personal memory of the period, there was one petition that really had a big impact. "Road pricing" was being discussed inside the government very seriously. In the middle of that a petition got started that attracted over 1 million signatures and the policy was dropped.