We don't have any 20 gauges, but I am partial to the Ithaca M37 Deerslayer with a 20 inch barrel. I also like the old Winchester Model 97. I'm not up on what the new gun market has to offer, but there is the Remington 870, as well as a Mossberg or an FN. The FN is based on the Winchester.
A pump or semi-auto?
Armscor markets a budget priced pump that I believe is based on the older Hi-Standard.
A hacksaw has made quite a few shotguns into home defense / riot guns over the years.
We have an old remington 870 with a folding stock, pistol grip and the seal of the delaware state police. It has an 18.50barrel and what i think is an 8 rd tube under the barrel. Had it for 20 years and finally took it to the range and shot 2.75 inch shells in 7, 4 and 00.even with the metal stock at my shoulder it was not unpleasant. I wouldn’t hesitate with a 20 (7/8 oz in a target load) either, but at room distance, I am thinking a light 12 ga load of 6or 7 ( 1 oz game load) is formidable and easy on recoil. I would go with a recognized brand, as said above, be sure its operational and get in osme range time.
My current closet shotgun is an 20" 870 12ga, but I just purchased a Win 1300 (police trade) 7 shot 12ga. 18" that may be taking its place. I don't think there is any reason to go with 20ga over 12ga. I keep mine loaded 2 rounds of bird shot, then buck. I also think a mossberg cruiser (the pistol grip model) is very maneuverable and loaded with birdshot would make an excellent home shotgun.
The local shops can't keep used Mossbergs which they cut to 20" for the customer. Don't use a load that may be as dangerous to your neighbors as an intruder. No. 6 shot is devastating at close range, and won't shoot through your house and the neighbors.
I have a Remington 870 in .20 gauge, and a couple Mossbergs in .12 gauge in the safe, but I keep an older Manhatten Arms SS in .12 gauge as my home defense shotgun. I had the barrels cut down and the bead reset, double triggers loaded with standard 2 3/4 inch 00 buck. Another O/U in .12 gauge with short barrels is my second home defense gauge. Extra shells ride on the stock in a shell carrier.
Mossberg 500 12 gauge pump. I like the placement of the safety on top of the receiver which puts it right under the thumb (left or right handed). The only accessories I would add are a nice bright light (500 lumens or more) and a cartridge holder. Five round mag tube is enough. Low recoil #4 buck is my preferred load.
This is a Mossbery/Cobray 500A " Rogue". Since it has an 11.75 inch barrel it is classified as "AOW" and transferred with a $5 fee. It's great for tight spaces, is easy to conceal, and fits in the dresser draw next to the bed; however, as a 12 ga it is hardly a fun gun to shoot. It's strictly an up close and personal type of firearm.
IMO, shotguns are not ideal for home defense. The only redeeming factor is the intimidating sound they make when you rack them, but I prefer something that can be held and fired with one hand. Just imagine trying to open a door or being on the phone with 911 while holding a shotgun in ready position, and you'll see what I mean.