The first "e-petitions" - which allow the public to prompt parliamentary debates if they get enough support - have been published by the government.
The return of the death penalty heads the list of demands, with dozens of separate campaigns calling for it.
However a petition opposing its return was the most popular one six hours after the site launched.
Other suggestions include keeping all Formula One races on terrestrial TV, leaving the EU and a lower voting age.
Petitions gaining more than 100,000 signatures could lead to a full debate in the House of Commons.
Ministers have warned MPs not to "ignore" the public's suggestions.
More than 40 of the first 200 or so petitions published called for the return of capital punishment, the most popular attracting more than 1,000 supporters by 1800 BST.
But the most popular petition was one opposed to the return of the death penalty, which had gained more than 2,700 signatures.
Several petitions call for all Formula One races to be shown live on terrestrial TV, following the announcement last week that the BBC will be sharing coverage with Sky Sports.
The most popular gained more than 1,000 signatories by 1800 BST.
One petition recommends the televising of court proceedings and another that the price of alcohol be increased.
One demands that prisoners' diets be restricted to bread and water, as in the "good old days", another that bodybuilding should be encouraged to improve the nation's health.
Any petition signed by more than 100,000 UK citizens goes to the cross-party Commons backbench business committee, which will decide whether it is worthy of debate.
This does not mean any parliamentary bills will be tabled as a result, simply that the matter will be discussed.
Leader of the House of Commons, Sir George Young, told BBC News the threshold for debating an e-petition could be raised if too many reached the signatures target.
He said: "We do want to monitor it to see if we've got the threshold either too high or too low.
"This is a new initiative and we've set 100,000 because we think that's roughly the right target, but if lots and lots of petitions sail through that barrier then we may need to see if it should be higher.
"If none of them are able to reach that target then we may need to lower it."
Since launching on Thursday, the e-petitions site has experienced problems, with about 1,000 people a minute logging on.
A government spokesman said: "We apologise for any inconvenience experienced as people try to access e-petitions - this is a result of greater-than-expected demand."
Any petition deemed to be libellous, offensive, duplicates of existing open petitions or not related to government will be rejected.
Moderators will also block any that concern honours and appointments.
But Labour have said the petitions could lead to debates on "crazy ideas".
The system replaces the previous e-petitions pages on the Downing Street website, set up when Tony Blair was PM.
The most popular of these, with more than 1.8 million people in support, opposed road pricing.
More than 70,000 backed the one-word suggestion that Gordon Brown should "resign".
And almost 50,000 signed up to the idea that TV presenter Jeremy Clarkson should become prime minister.