So what's the typical maximum IAS seen on a typical 737 flight, then?
They have an Mmo of 340 knots, but I doubt they usually get anywhere near it.

Never flown a 737, but they are probably at or near Mmo during descents and in cruise at lower alititudes.
There are several airline pilots that are also glider pilots, so maybe they will reply.
Although, they may be too busy for ras.

If it helps, the IAS for a C560 at FL410 is usually around 230 kias which is ~420 true.
At FL350 the indicated is 250 and below FL250 the power must be reduced to stay below Vne of 290 even in cruise.
Descents from high altitude can only be at full power for a few minutes before Vne is reached and typical descents are done at reduced power.
If allowed by ATC to descend at pilot discretion, then it is easy to descend at Vne with reduced power to the assigned altitude.
If an expedited descent or agressive crossing restriction is given, then the descent may be at idle and sometimes even with spoilers out to stay below Vne.
Obviously the lower the aircraft is (i.e. closer to standard sea level pressure and temperature) the closer the indicated airspeeds is to the true airspeed.
It is usually necessary to start slowing down by 15,000 to be under the 250 knot speed limit at 10,000.
290 kias at 15,000 is 380 knots and 250 kias at 10,000 is 300 knots.