I've also had some luck with a combo of those magic erasers & baking soda.

Yep. Bartender's Friend is exactly what's recommended for pans like that, even by the manufacturer.

BKF is great but my only complaint is that you really have to rinse your sink out after you use it. It just sticks to my stainless sink.

If your pan is seriously yucky, plain old SOS pads - those soapy steel wool pads, work wonders.

For really stuck-on stuff, I sometimes will scrub some coarse salt around with a tiny amount of water. Depending on how bad and how "stuck" it is, that may or may not work - but I think the other commenters are right on. Give it a shot!

My first strategy is usually to boil an inch or so of water with a big squirt of dish soap for 10 minutes. This helps to lift off particles and gunk. Then I let it cool, and scrub away. After that, I'd do Barkeeper's Friend. Like others said, it is recommended by many stainless steel cookware manufacturers and it does not scratch! Make sure you wear gloves when you use it, it does weird things to your hands.

I use the much more abrasive Comet cleanser when my pans are really bad. They never go back to that mirror-like finish, but they sure do get clean.

I make a slurry of baking soda and water to cover the bottom of the pan, then boil on the stove until it gets paste-thick. This seems to lift off the black gunk, though you still have to scrub afterwards.

I have a lot of luck with baking soda, I'll fill the pan with water and dump in a fair amount of baking soda and let it sit. then scrub

I went through all sorts of different methods myself, including BKF, vinegar, boiling with dish soap and elbow grease. Finally, the easiest method of all, which is baking soda with just enough water to make a dryish paste. It works so much better than anything else I've ever tried.

Ammonia works really well on my pots, pans and racks. Try placing the pots or racks in a garbage bag with a cup or so of ammonia. And let it sit outdoors for about 24 or more hours. After that time, the burnt spots should flake right off.

This tip is from eHow--

1
Place the oil stained pots and pans into a large trash bag. Carry the bag to an outdoor location.

2
Open the mouth of the bag and pour 1 to 2 cups of pure ammonia into the bag. Try to concentrate the ammonia in the location of the stains. If the stains are inside the pots, pour the ammonia inside the pots. If the stains are on the bottom of the pots pour the ammonia into the bottom of the bag so the pots are sitting in it.

3
Close the mouth of the bag tightly using a twist tie or tape, and place the bag in a sunny location. Allow the bag to sit for approximately 4 to 8 hours.

4
Open the mouth of the bag. Do this quickly, and then walk away from the bag to prevent inhaling the ammonia fumes. Wait 5 to 10 minutes for the fumes to dissipate.

5
Put on gloves and remove the pots from the trash bag. Spray the pots thoroughly using a garden hose to remove most of the ammonia from them.

6
Take the pots inside and wash them immediately with hot water, dish soap and a scrubber sponge to remove all final traces of the oil stain as well as the ammonia residue.

7
Dry the pans thoroughly using a cloth.

Tips & Warnings
Do not use this method on anodized aluminum pots that have a nonstick surface.

There's a few scrubs including Bar Keeper's Friend. You can also use Zuds, and I've used Bon Ami on my stainless cookware. I use it with the magic eraser sponge, which is a great pair. I used to let my pots dry by themselves, but recently I've started drying them immediately after washing. I boil a lot of water to make iced tea and have been getting rust pitting.

I once boiled away a saucepan of water, causing the metal to bubble on the inside bottom of the pan. I poured a 1/2 can of a specific soda (rhymes with Joke) into the pan, and the next morning wiped it clean. It was a combination of frightening and impressive; my pan looked like new.

My pans when I bought them instructed to use BKF or a generous sprinkle of powdered dishwasher detergent just covered with water and boil that. For lots of cooked on stains, that works best for me but BKF is awesome.

Yeppers, BKF... And for oven racks, soak overnight in a bathtub (NOT the acrylic ones...) with dishwasher detergent powder. If the gunk is really baked on, you might have to use a stainless scrubby but most of the magic is already done!

VINEGAR + BAKING SODA

Been using the ammonia + garbage bag trick all my life. Caution: if you do this, be sure to open the bag away from your face:)

Cameo (a brand name) Aluminum and Stainless Steel powder. It comes in a plastic shaker can. Cameo also makes the best copper cleaner/polish on the market. Nota Bene: the aluminum powder will take pot marks out of porcelain sinks---it's practically a miracle.

I use cheap (2/$1) long-handled scrub brushes from Ikea for most hand washing. They seem to work better than dishrags and sponges for cleaning pans. They also add a little extra oomph to some of the methods above.

If you want to avoid any type of cleaning product (even Barkeeper's Friend), I second Operator, vinegar and baking soda work great. I've used this method in the past when the stains were just on the bottom of my pot: Pour white vinegar to completely cover the bottom of the pot and bring it to a simmer, remove from heat and add a handful of baking soda. Let the vinegar baking soda slush sit for 10 min and then scrub with a scouring pad. For your roasting pan/rack, I'm sure you could bring the vinegar to a simmer in your oven.

Bar Keeper's Friend is the only game in town when it comes to cleaning stainless steel. :)

Cooking on a commercial range kills pots and pans. Like others I use BKF all the time.
But when that does not work I swear by Dawn Power Dissolver. Its sometimes difficult to find or if a store carries it they seem to sell out of it.
It's a spray on product that smells like oven cleaner -which it probably is and can be left on to soak. There is hardly any scrubbing involved.
I have had it work on old baking pans too.

I use the same method as PEACHY44. If you regularly do some vinegar/soda maintenance, your pans will stay looking brand new.

BKF is usually easy to find, but Bon Ami is a superior product. They both do the same thing and work wonderfully, but I appreciate the non-toxic aspect of Bon Ami.

I know I've mentioned previously how much I like Bon Ami but clearly I'm a fan.

I always have success by boiling some water and a Method dishwasher tablet (Smarty Dish) in a difficult pan.

I usually scrub with a baking soda paste like many people suggested above.
For really hardcore jobs, I use diluted chlorine bleach that I let sit in the pan for 24 hours, then I pour it out, rinse very well and wash thoroughly with regular dish soap. This works wonders. Just make sure to rinse very well.

brillo pads are AMAZING and I don't find that they damage my pots. Another suggestion is to soak the pots with very hot water and a few tablespoons of powdered dishwashing detergent. Works wonders for getting stuck on food to come right off.

Bar Keepers Friend does, indeed, work well on stains on stainless steel - the oxalic acid does the magic. However, it is very important to be careful with it. We have dark laminate countertops, and the BKF bleached little spots on our counters - rinse, rinse, rinse!

Bak Keeper's Friend!

The easiest, cheapest and greenest method is simmering baking soda in an inch of water in the pan for about 15 minutes. Either let it cool slightly or attend to it later in the day. Use a scrubbie to clean the loosened char. Baking soda paste works great on Corning ware, Formica, glass, etc. Why use caustic, toxic, chemical-laden products, especially in your cookware?

The nicest on the environment (including my hands) that I have ever (and so continue to) use is a product by SHAKLEE CORPORATION called SCOUR OFF. It smells like bubble gum but cleans my stainless until it looks new! I adore it :)

I haven't tried bkf. But will try this out. I found boiling with dish soap not to do the trick. But I use cerama bryte (glass ceramic cooktop cleaner) and a dobie. You never want to use anything abrasive or anything with bleach on your pots and pans. Remember to go with the groves As it it easier to clean. Just put the cerama bryte in your dry pan and use your dobie and go in a circle or however your groves go. You can also use it in your stainless steel sinks and won't leave that dull white film behind. I've also use it to get the color spots of the inside bottom of my pot. I've also used it on the Bottom of the pans. (The part that touches the burner) I refuse to use comet as this leaves a film behind in my sinks. You will find this method very useful and very satisfying. With no residue left behind! Best of luck.

There is a Dutch company that has a very nice solution for this. It is even used in medical environment. I think they've retailers all over the world: Stubborn stains from stainless steel.