Supposed I'm a middle-class man in Early Renaissance London and I wanted to carry a weapon on my person in public. What sort of rules and regulations were there for me to own and carry one?

You would not be allowed to carry a sword in London. Cities in France and England generally had laws restricting carrying a sword to only certain classes. In particular, carrying a sword and buckler at night was seen as a heinous crime. We have lists of names of people arrested for doing so. The earliest example I could find is from the late 1200s:

Henry de Wynchester, Michael de "Womburne," Oliver the Goldsmith (le Orfevere), Walter de Molesham, Richard Bonaventure, Robert Pynnote, William de Clay, Nigel Lupus, John Fuatard, "barbur," Thomasin le Barber, Alan de Schoresdich, John Aufre, Edward de Canterbury, goldsmith, Thomas le Fykes, Alan de Suffolk, taverner, Reginald, son of Emma le Barbere, Roger the Cook, "pasteler," Reginald le Taverner, James le Reve, "pessoner," Walter de Resslepe, John Squiret, Peter le Tableter, John, son of Roger le Barber, William Stonhard, Ranulph le Taverner de Wolcherchawe, Laurence "Ballok," John le Fevere, Adam de Wynchester, John le Treiere, John Burnel, and Alan de Ewelle—arrested for divers trespasses, homicides, robberies, and assaults, and for being nightwalkers after curfew in the City with swords and bucklers, and for setting up games near the City, contrary to the King's peace and the ordinance and statutes of the City of London— say that they are guilty of none of these things, and as to this, each puts himself upon the verdict of four jurors of each Ward of the City aforesaid. Accordingly let there be an inquest thereon.

http://www.british-history.ac.uk/london-letter-books/volb/pp1-15

More from the late 1200s/early 1300s:

Bartholomew de Hallynge, John Fuatard, Nicholas le Quelter, Robert le Ster, Andrew de Kent, Roger le Avener de Garscherche, Richard Fretemone, Robert le Bere, Thomas de Arderne, John le Treere, Ralph le Treere, John Jyvenal, Alan le Mazun, Fulk le Barber, Simon de Enefeld, Peter le Tabellion, (fn. 24) John de Benetlegh, Roger le Lechiere, Peter de Honilane, William le Coffrer, Walter de Danecastre, John Barber Picard [sic], Walter le Veyre de Ospreng, Hugh le Barber, "William" le Taillur de Colem[anstreet], Robert de Lincoln, Adam Pikeman, fishmonger, Adam Pikeman, butcher, and John, son of William de Enef[eld]—arrested for divers trespasses, as for homicides, robberies, beatings, assaults, and for being vagrants by night after curfew in the City with swords and bucklers, and for instituting games near the City and keeping houses of ill fame, contrary to the peace of the lord the King, and contrary to the ordinance and provision of good men of the City aforesaid — say that they are in no way guilty of the trespasses aforesaid. And touching this, each for himself puts himself on the verdict of four jurors of each Ward of the City. Let there be an inquest accordingly.

Here's one from 1416:

Another writ to the same to the effect that whereas the King of the Romans was now approaching the City with a noble retinue, and it was King Henry's wish that his lords and other of his lieges from divers parts of the realm should come and reside in London during the Emperor's stay, the Sheriffs should make public proclamation restricting the carrying of swords or other weapons in the City to knights and esquires under penalty of forfeiture.

http://www.british-history.ac.uk/london-letter-books/voli/pp157-166

It's important to note that these restrictions applied to towns and cities. As swords became less expensive, they became popular self defense weapons for travelers of most classes. Sword and buckler are ideally suited self defense weapons for travelling. A sword can be carried as a sidearm, and a buckler can hung off the waist.

tldr; In a city you would have to either be a criminal or Knight/Nobleman to carry a sword. While weapons laws were much more relaxed for those outside the city.