What is the widest tire you can mount on a old school 27 x 1 1/4 rim

Generally 1.5" maybe a bit more...depends on what will fit in the fork and rear stays...also smoother tires would be better than a knobby tire...the 'knobs' tend to make the tire wider than it really is.
I generally always use a 1.5" up to 1.75" tire in the shop when replacing tires on the size wheel you mention.

First and most important:

Determine whether the inside of the rims is straight or has hooks at the outer edge to retain the tire. Straight-sided rims limit the maximum safe pressure. Having seen tires blow off straight-sided rims, I ignore the "inflate to" instruction on the tire and pump them to no more than 65 lbs.

Second:

The widest tires currently available to fit 27 x 1 1/4" rimes are almost certainly 27 x 1 3/8". Among the few tire models available in that size, some actually measure only a small fraction of an inch wider than the standard 27 x 1 1/4" tire. Panaracer and Continental tires are usually mentioned as measuring a true 27 x 1 3/8".

27.5 is the newer name for 650B and has the same 584-mm bead seat diameter. 27.5 rims can fit the tires you mentioned. The OP has 27 x 1 1/4" wheels: 630 BSD.

Thread moved from Introductions to Bike Mechanics.

The sizes 27 1/2" and 27.5" are not interchangeable.

Keifer97, the widest tires made for 27 inch rims are 1-1/4 and 1-3/8.

With an actual 27x1-1/4" ISO 32-630 there's generally not much choice, some 27" tyres are effectively 27x1-3/8" ISO 35-630 (used to be you could get narrower 27" tyres, down to 7/8", but not wider). Switch to a 700C rim which is 4 mm smaller radius and you have a much wider choice of tyres, but with anything wider than 35 mm you'll likely start to have clearance problems, usually between the chain stays.

Yep...that is what I read and know the diff between 27.5 and 27 1/4.
I install 1.5" wide tires on 27 1/4 wheels without a problem in general. They are readily available at two of my suppliers. I also know the diff between the BSD's...not all 27 1/4's are alike.
Typically we replace the tires with the standard 27 1/4 tire but we do get asked for a wider tire and if it fits we use it.

What specific tires are 1-1/2 inches wide for 27 inch rims (630 mm diameter)?

OP, see this thread for full coverage of the 27” options. The ultimate 27" tire reference thread!

Yes, I'm puzzled about this, too, and would like to see some examples cited. It's possible that one or two modern tire manufacturers label their models of wide 27"/630 tires 1.5" rather than 1 1/2", but that would be unfortunate. Referring to a 27"/630 tire size labeled 1 1/2" as 1.5" perpetuates the confusion that Sheldon Brown disambiguated by presenting his tire size information in two separate tables as "Fractional" and "Decimal."

When I'm in the shop I'll take some pics

Or just link to online catalogs showing the 27 x 1.5" and 1.75" tires that you mentioned.

Edit: just did a quick search of websites of online suppliers for 27 x 1.5" and 1.75" tires. Found none.

That list was good in its day but you'll be hard pressed to find most of the covers it lists in 27" nowadays. Still available are Schwalbe Marathon and Road Cruiser, Michelin Protek, Continental Ride Tour and Super Sport, Panaracer Pasela, all 32-630 And of course various no-name tyres - do you really want to go lower than Schwalbe Road Cruiser?
Any others?

And then you might run into problems making your brake calipers reach. But, hey, you won't know til you try.

not going to do that...these are our wholesale vendors...

Yes, I checked QBP and saw that they apparently no longer maintain an online catalog (without prices) for consumers to explore. Would you mind providing those shop photos of the tires?

Either that, or links to online retailers of the same tires. Seems unlikely that such tires would be made available to bike shops but not, e.g., Jensen. (Just checked: I found a couple of dozen wide 27.5 tires on the Jensen website but no 27" tires of any width, let alone any 27" tires with a width designated by a decimal.)

When I'm at the shop...off season now so no work for me until spring but I'm usually in the shop once a week to hang and chat...I'll see what we still have in stock and look at what we can still get online.