I've finished dry-fitting a new branch (6-inch round) from the main furnace duct to a new register. I've supported each section of duct with metal strap and have begun taping all connections with foil tape. The tape and strap seems quite sufficient to hold everything together. Are screws required for every junction? The other branches in this (older) home are screws-only; no tape.
What I've learned thru doing residential HVAC for a couple of years in Michigan is that for it to be up to code you need at least three screws for under 8 inch diameter ducts. Anything larger deserves at least four. Keep in mind it's not always possible to evenly distribute the screws, so this is why you also want to use hangers of some sort.
Finally, use mastic sealant or aluminum tape; be sure to degrease your duct when using tape as the duct has oil on it from production and tape will not always stick.
Yes, now a days screws and a sealant of some sort are required. Use aluminum tape, not the gray plastic crap with cloth backing, it falls apart in a few years. There are brush on sealants also.
Self Screwing Sheet Metal Screws
Air Duct Sealant/Caulk
Aluminum Foil Tape
Duct Wrap (if in unconditioned space)
An alternative to a screw is to use a rivet like a pop rivet. You need a diamond hole cutter to make the hole for the rivet.
Using rivets will result in a tight, permanent connection that will not work loose and will not have any sharp protrusions.