I'll second the J&C Pro 100 -- very nice B&W film, in some ways the equal of Plus-X. Keeping the temperature carefully at or below 68F/20C, and avoiding the acid stop bath seem to help tremendously in avoiding bubbles and pinholes, though you'll still find very occasional coating defects (unavoidable, says John from J&C, due to the WWII vintage coating equipment in use). If you're shooting something critical, it's worth shooting two rolls; if you have defects, you'll probably have them spread along a whole roll, but it's very unlikely a second roll will have them also.
I've been shooting a lot of Pro 100 lately -- and I really like the black plastic film cans that are included in the price.
My other main 120 B&W is TMY. A lot of folks don't seem to like it, but in my process and to my eye, it's hardly any grainier than Pro 100, gives a full 400 speed, and has very nice gradations and tones. This might have to do with the fact I scan my negatives (until such time as I can get my bathroom, er, darkroom set up to print), but prints I've gotten from TMY have always looked wonderful, too.
Since you're on a budget, however, you might try Lucky SHD400. I've recently shot and processed a roll; I found it to look a good deal like Tri-X, though the base is very thin and the film likes to curl into a tube after drying. Some have complained the SHD films have poor toe speed, but I didn't notice a problem in my process (dilute developer, low agitation, and a long time in the soup), which tends to pull up the toe. The price of the SHD is very nice, too.
Another good budget option for faster film is Classic 400 -- this is Forte film rebranded, and though I haven't tried any, I've seen pictures from it that looked very good.
FWIW, in case you have trouble with X-ray damage with APO shipped film (and you might), all of these except the Pro 100 are available from fotoimpex.de and from Retrophoto in the UK; shipping within the EU shouldn't be a big deal, but the VAT can make you crazy...