Does divorce occur if the wife wears the niqab at home, or if she goes out into the street with it after her husband swore an oath of divorce upon her that she not go out into the street with it?

If you swore to your wife by divorce, intending to prevent her from going out into the street wearing the niqab, then you do not break your oath if she wears it inside the house or in a place other than the street. However, if she goes out into the street wearing it, the majority of scholars hold that the divorce takes effect. Some, like Ibn Taymiyyah, are of the view that it does not take effect if your intention was to threaten or emphasize, without intending divorce, and in this case, an expiation for a broken oath (kaffarat yameen) is due upon you. It must be noted that the permissible oath is by Allah Almighty, whereas swearing by divorce is blameworthy and may lead to serious consequences.