Lady Blue being in approximately the same situation as you has bought a lot of her bras from Asda's George range.

Most of the ones she has bought have been basically as you describe - "boxed, cheap, dead-plain bras. They won't be exciting, or available in anything other than white, but they will be functional". Except Asda do them in black as well.

They are also stupidly cheap. Like £2 each or summat.

As the person who often does the washing they have been going through the wash for a year or so and appear to be still un-disintegrated.