Can anyone tell me what the distinction is between the above 2 terms for 'laundry'? Is one of them 'self-serve' (laundromat)..? The online translation tools give same result for both. Thanks.

The German word die Wäsche is a generic word for laundry. No plural.
It simply means dirty clothes that need washing. Ich habe eine grosse schmutzige Wäsche. means I have a lot of dirty laundry.
die Wäscherei is the establishment -- the laundry business. There is no automatic connotation that it is a coin operated self serve laundry.

Der Waschsalon is perhaps a laundromat in our definition. The dictionary I have translates it that way.

We need a resident of Germany to tell us more about the availability of self service laundries in Germany. I am sure there are regional differences.

I have heard the following phrase used to denote a laundromat:

Eine Wäscherei mit Automat.

That sounds like one to me!!
A Wäscherei is a place with the equipment to wash clothes. One of the definitions of an the term Automat is a slot machine or a vending machine.

Cool..! good to know in case we need to be 'kind' to fellow travelers... Thank you.

In the five months we spent in Germany (in the Ruhr area) this winter, we found that some self-service places called themselves Wascherei and other Waschsalon. A waschsalon is virtually always self-service; the wascherei usually is, but occasionally may be a drop off laundry.

Doesn't Waesche also refer to lingerie?

Die Wäsche is used in my experience to mean general washing. Die Damenwäsche can refer to various women's undergarments.

My guess is that quite a few regional slang terms exist that change the meanings of ordinary words.

A good example of that is the word "gay" in English. It used to mean happy and joyful.

Well, this sounds like fun to a German

But pretty much all of you are right:
My own dictionary says Waesche (with the Umlaut) means the act/ procedure of washing; the things that need washing; undergarments.

I personally think of laundry first then of underwear (both male and female, by the way).
Waescherei is a term I rarely hear and read these days. Since laundromats with the machines that you operate yourselve are normally called Waschsalon and the laundry drop off places, as well as dry cleaners, are called Reinigung.

And jemandem an die Wäsche gehen means - well - to get into someones pants ;-)

now who would think a question about something so 'clean' could meander into 'dirty' thoughts..?!

Good one Bob!

"Damenwäsche" means just that, washing ladies.

The word you are after is "Damenunterwäsche" which is ladies' underwear.

Hahaharzer

To add some more variety to this question, I just came across my receipt for laundry from Zermatt. Items were listed under "Damenwasche, Herrenwaschen or Kinderwaschen." Interestingly, unlike many dry cleaners in the US, men's and women's clothes cost the same, only childrens were less.