For uses not pertaining to childbirth, use borne—grudges, burdens, and brunts are all borne. For childbirth, both borne and born can be correct depending on the phrasing: use borne in the active (I have borne no child, She has borne two) and in the passive when preceding by (Only the two borne by her have died); use born in the passive when not preceding by (I was born here, Three were born to her) and in the adjectival sense (He was a German-born musician). Both words are past participles of bear.
The set phrase born and bred is so spelled.