If someone absolutely loves their fridge please say the model number! These things are all over the place in reviews.
The fridge we got:
https://www.kitchenaid.com/shop/-[KRFC300ESS]-408402/KRFC300ESS/
Note cold water dispenser inside on the left.
Why counter depth? Can’t open the doors and stand in front of it? You pay a lot more for less storage. Even the so called counter depth ones will not be even with the counter. Function is so much more important than a smooth line along the wall.
I chose a Samsung French door refrigerator a couple of years ago. 28 cubic feet for two of us (plus the old 5 cubic foot chest freezer brought with us). LG and Samsung were the two with freezer doors that lift up a bit so your pulling motion is nicer (check it out in the stores). A salesman told me about differences in ice production between those two brands. Reliability is another biggie- I got the LG washer/dryer front loaders for that reason. Energy efficiency labels changed around then but by now they are now the more realistic ones.
My Samsung. I like the ability to change a half shelf to store more tall items occasionally and other details. No refrigerator is perfect- some of those on door storage spaces are unusable… Definitely like the French doors. So much easier accessing fruit et al bins without bending. Be sure to get solid, not wire, freezer shelves so stuff can’t fall through (that partial bag of peas could become open…).
" Function is so much more important than a smooth line along the wall."
Yes, it is all about function - counter depth fridges can be located where it is convenient to have them without blocking the isle. A regular fridge will stick out up to 10 inches out for the doors to be able to open normally if you have cabinetry on both sides. I hate getting bruises when I run into corners. Counter depth or built in for me, please. 
I bought this LG counter depth fridge a year ago and love it. http://www.homedepot.com/p/LG-Electronics-22-7-cu-ft-French-Door-Refrigerator-in-Stainless-Steel-Counter-Depth-LMXC23746S/205346627 The middle drawer has a flexible temperature setting. They make the same version with a see through right door.
I liked shopping the Home Depot website because it allows you to choose based on height, depth, color and several other criteria. I did purchase the actual fridge in person because I like to see and touch it. There are only 2 of us and it is more than big enough.
We need counter depth because of the placement of our fridge…in sort of a walk through between our island and the fridge and pantry cabinet. If we didn’t have a counter depth fridge, it would stick out about nine inches into the calling area…and would look even worse than the 3 inches or so that it does stick out.
It was MY fault…if I had realized this…I would have moved that wall behind the fridge back six or eight inches…so I could have gotten a regular size fridge…and it would still have been flush with the cabinet front. I don’t need a flyer as big as I have!
I have a small 15-cu ft counter depth fridge. My 22-year old Hotpoint needed to be replaced so I measured it. Sears didn’t have the size so I checked online and had no luck. Then I called and learned that GE had bought out Hotpoint. They had a similar sized fridge and there was one at our Lowe’s. Lowe’s didn’t have that model when I checked there earlier.
I wish mine would make ice or have a bottom or side freezer. Guess I’ll need either a bigger house or a bigger kitchen.
We did what Thumper suggested - the space for the refrigerator is deeper than the refrigerator. We have a regular sized refrigerator and its pretty flush with the surround cabinetry. Looks counter depth, but its really much deeper. I didn’t want to sacrifice the additional space with a counter depth refrigerator. We have some sort of Kitchenaid french door model. Its 7 years old so I’m not sure they still make it. Its laid out just like the one Bunsen Burner linked which works quite well for me.
@Onward: From your description I think it’s the unit we just purchased several months ago. Purchased thru Costco. By using my Costco credit card it extended the ‘bumper to bumper’ warranty to 4 years. I like that particular model because it does not have protruding handles. These are flush with the doors. Ours sits on one side of the island and I didn’t want to lose floor/maneuvering space by having handles stick out several inches.
So far we like the unit. We’ve configured the lower compartment for fruits and veggies. Since we have two standup freezers (H fishes A LOT) we don’t need the extra freezer space. The only complaint so far is the the grayish stainless steel finish is a finger print magnet.
We had a 17 year old side by side Kitchenaid with water and ice in the door. It was still going strong.
If you don’t have extra space for depth, try to have a recessed outlet put in, like @coralbrook does, and you can gain 3" or so for the big refrigerator plug.
As long as the water hookup is also recessed… if you have water/ice maker in your fridge.
And as long as there are no requirements for air flow.
I have my fridge in a big built in shelving unit. It cost almost as much as all the rest of my cabinets! (In my avatar.) It’s a big fridge, but I hate it - side by side that refuses to die. I just hope I’ll find something the same size when it comes time to replace it.
We were in a similar situation. If your fridge is a standard fridge, you will be able to find a replacement. Your options might be limited to a handful of models though.
Have a french door counter depth KitchenAid that does not have in door ice and water. This was one of their 1st FD models and was before they added in door models. Given the amount of storage space lost with those in door features, I’m glad I don’t have them. But then we only had 1 kid trying to get ice and water, not a large group constantly opening and closing fridge doors. Am scared I’m going to jinx myself but we’ve gotten more than what is considered normal life out of this one. I’m really happy with it, having the width of both doors is a godsend for us. Being short I can’t imagine a refrigerator that would “fit” me better. The freezer drawer is on the smaller wide but very usable and we have a small upright freezer in the basement.
As for finding a replacement, I’d talk to salespeople at big box stores about which lines have the most repairs. I’ve found employees at places like Lowes quite happy to talk about this if you can catch them at a slow time - like weekday mornings. The good ones know their product lines and don’t like to deal with selling junky appliances - more work for them to sell the good ones as people tend to paint appliances with a broad brush. Most gave me specifics about the problems like one manufacturer was bad at supplying parts, a certain poorly designed part was difficult to get to in the unit, etc.
When we replaced our fridge, we went to a Sears Allpince store. They had a good selection…but really…if I had it to do again…I would go to the local appliance store which has FAR better customer service, and also a large variety of choices (that’s where we got our stove).
So…check around you for locally owned appliance stores.
Local appliance stores will gladly match the big box prices, as we found, and as thumper said, the service will be much better, especially delivery - most employ their delivery folks instead of outsourcing to a third party. Lowe’s appliance sales folks are usually very knowledgeable, but the delivery was awful.
Our local store appliances cost a little more…but they dont nickel and dime you to death with additional charges. Free delivery, free haul away of old appliance, free cords (yes…Sears was going to charge us for the stove cord), so at the end of the transaction…the local store was actually less costly.
Our fridge was $100 less at Albert Lee than the advertised sale prices at big boxes (with all of their discounts). It never hurts to ask… Yes to the cords and hoses. We asked if we needed to buy a water connection for the fridge - it was a part of the free delivery. The best part - and we bought enough appliances for 5 deliveries - all delivery folks called me 30 min prior to arrival and all sowed up within the given 2 hour window. Only once the delivery was late - the customer before us did not bother to measure the front door… oops. The delivery guys had to take the doors off the hinges and take the doors off both the old and the new fridge in that house. It took them 2 hours to do what normally takes 20 min. Long story short… make sure what you buy fits through the front door!
My plan is to purchase either at Costco for the extended warranty, or our local independent appliance store. While I have purchased a few appliances from HD and Lowe’s for my rental home due to availability, and from BrandSmart for my red washer/dryer, I do the rest of my business from the independent store; they have handled my repairs for 30 years, not that I have had many.
My appliance store will match any big box prices, so I look at those stores first to see all the different washer /dryer and refrigerators lined up in one place. The showroom at my appliance store is huge as they do a lot of designing, but they do not have every appliance out on display; just their high end and best sellers in the low to med range.
Up north water from the refrigerator was not wanted but now here in Florida that 70+ degree tap cold water is much too warm compared to up north 50 degree groundwater.
I was able to get lowest prices I’ve seen even two years later for my appliance purchases from a close by hhGregg- that chain folded recently. Thank goodness I do not need any more appliances for a long time! Installation quality can vary, another consideration for washers, dryers and dishwashers.
Most refrigerators, regardless of freezer placement, tend to fit the same space- all less than 72" high and 36" wide for large standard types. Smaller ones fit fine.Of course, years ago some builder may have fit in a small model and space. Ours is at the end of a cabinet run, with the kitchen entry next to it (an always open sliding door). Cabinet run goes above it. Would never want to enclose it- let the refrigerator side show.
We got lucky with space in this house. Willing to have a so-so kitchen. Rejected one house with too much kitchen- let a family who wants/needs the fancier stuff. Made sure old house we had designed/built would accommodate the large versions of things.