First, there are two main types of champagne bottles that you find. Most common in the 750ml size.
26mm Crown closure
29mm Crown closure
The difference is minimal and I doubt you can see it in this photo but here's a corked 26mm bottle (on right) and a corked 29mm bottle (on left).
Here are the two different caps. Left is your 26mm standard crown and right is your 29mm crown.
The 26mm takes the standard caps we use on beer bottles. They are zero issue to cap with a bench capper and a wing capper can be used with little to no issue as well.
The 29mm requires the larger 29mm caps which seem to be a special order item usually. I have only ever bought one pack and it came with 100 caps. In order to use these caps and bottles you need to replace the bell on your capper to use 29mm caps. It's about $5 or less for the part and is very easy to use on a bench capper. Using a wing capper on 29mm bottles can be done, success may vary based on the size of the bottle neck.
In the 750mL size not all bottles are created equal. There are some bottles like used by Dogfish Head that are made just for putting beer in them. Many of them are only rated up to like 3.5/4 volumes. Not that any of us need to go higher. I'm just saying. Typically the bottles with a punt are heavier. The real champagne bottles are significantly heavier and stronger than the kind DFH uses. For reference check out this United Bottles link. You'll notice that the "Amber Spark" bottle is 650 grams when a "Heavy Sparkling" bottle is 900 grams.
The different champagne bottle sizes are:
375mL 29mm Crown (100% the case in my experience)
750mL 26mm and 29mm Crown
1.5L 29mm (most likely, I only have one but 26mm might exist)
Uncommon sizes: All, but one, cork only to my knowledge.
3L (can be had with flip top) Here
Photo Caption: From left to right. 9L Salmanazar, 6L Methuselah, 3L Jeroboam/Double Magnum, 1.5L Magnum, 750mL, 375mL.
This image shows two different 3L champagne bottles. Apparently these bottles can accept a crown cap, but I've never seen a cap that large. I am still trying to track down a place that offers supplies for packaging in these bottles. I'd sure love to get the proper cages and stuff so I didn't have to make them.
Beyond 12L and I don't think a homebrewer would touch the bottle. I have not even really seen a 9L so the chances of a homebrewer getting their hands on a 12L is highly unlikely (though I'm trying). The larger ones can be slightly spendy. However buying an empty used one for $30 is better than paying $2000+ for a full one. These 3L and larger bottles are all named after biblical kings for whatever reason.
The "Belgian" bottles only come in two sizes. You have the standard 750ml size which Ommegang, Duvel, Chimay and many others use. Then there's the smaller "mini" Belgian bottle that Russian River, Port Brewing, and Lost Abbey utilize. These bottles MUST be corked.
Here's a picture showing the two bottle mouths of the standard Belgian bottle and the crown version of the same bottle.
An oddball of a bottle are the "Belgian Crown" bottles. They are Belgian shaped but instead of the "bullnose" at the top there's a crown cap closure. These bottles are used by Jolly Pumpkin and a few others I'm sure. They are lighter still but should have no problems taking the pressures we'd throw at them. Zero issues for a bench capper and should not give any problems for someone capping with a wing capper.
Here's a bottle I've had a while but haven't packaged yet. This is a prosecco bottle. I think Saison Voisin can be had in a bottle this shape as well. They come two ways, one that must be corked like this one, and one that accepts a 29mm crown. I have seen them packaged with standard wine corks then in place of a cage it had a small disk and strong twine.
When it comes to corking these bottles I use a Ferrari champagne floor corker. For the bottle sizes of 375mL - 1.5L I use 25.5mm Belgian corks in both champagne and Belgian bottles with zero issues. When I was in the market for a corker I figured I'd just drop the extra $$ and not work on a way to figure out how to making another piece of equipment work when I could buy one that already worked. I know you can use some other corkers to bottle standard Belgian sized corks.
Two walkthroughs using the Colona corker/capper.
Brew Your Own: The How-To Homebrew Beer Magazine - Beer Styles - Corking Belgians
SLO Brewer » Corking Belgians
Here's a walk through for using a Portuguese corker.
As a final note if you ever find yourself with a 3L or larger bottle and end up scratching your head trying to figure out how to cork it, here's a solution. First off, the cork hole is ever so slightly larger than a standard champagne opening (might be the same size but it seems slightly larger to me). So for a beer I think a standard champagne cork works just fine. What you need to do, unless bottling more than one, is compress the cork jaws on your champagne cork 2-3 hours before you plan on bottling. Get your large format bottle cleaned sanitized and filled. Then you quickly pull the cork out of the jaws, place it in the bottle, and wait for it to expand. It doesn't take long but you don't have to be super fast either. Afterall if you screw up you have a 3L+ bottle with beer and no way to close it up so steadiness is key. Additionally you will need to have a restraint cage made out of wire fashioned up yourself. I have not been able to figure out where the cages for these larger bottles can be obtained. The other option for closing these large format bottles would be permanent modification to a floor corker, or possibly using the bench champagne corker with some ingenuity. I think a bench corker could be easily modified to have a bottle stick through the bench you stick the corker on allowing for non-permanent modification.
Here's my first "homemade" wire cage for a 3L bottle.
I will add pictures to this thread as I take them myself.
I am also soliciting information for using plastic corks because that allows some to use the Belgian bottles with out having a corker.