Breachis a noun referring to (1) an opening or gap or (2) a violation or disruption, and a verb meaning (3) to make a hole or gap in or to break through. Breech is only a noun. It refers to to (1) the lower rear portion of the human trunk, or (2) the part of a firearm behind the barrel. The term breech birth (meaning the feet- or buttocks-first delivery of a baby) employs the word in its first sense.
Broach means (1) to make a hole in something, usually to draw off liquid, or (2) to bring up for discussion. The first sense is rare and not to be confused with the verb sense of breach. To breach is to break through; to broach is to make a hole in something for the specific purpose of drawing off liquid.
These writers use breach correctly:
A shipment of Iranian weapons seized in Nigeria didn’t breach any law, according to a lawyer representing an Iranian national charged with illegally importing them. [Bloomberg]
Two lawmakers have urged the FCC to conduct a full investigation into a privacy breach involving Google’s Street View. [The Hill]
As it breached the surface, the sub’s rudder sliced the hull of the Ehime Maru … [The Japan Times]
Examples of breech used in accordance with its conventional senses are rare on the web. Here are two of the few we found:
It bore no more weight than my older brother’s breech birth and my younger brother’s birth via C-section. [San Diego Reader]
And these writers use broach correctly:
A recent letter in your column about disclosing a past relationship has prompted me to finally broach this sensitive topic. [letter to Washington Post]
I hesitated, as I knew I was about to broach a touchy subject. [Los Angeles Times]