Facebook Inc. faces 10 investigations by Ireland’s privacy regulator into whether the company or its subsidiaries have violated European Union privacy law, making the social network the biggest target for one of the bloc’s most important data watchdogs, amid growing scrutiny around the world of its privacy practices.

Some of the investigations, disclosed as part of the Irish regulator’s annual report Thursday, focus on whether the company is legally gathering and processing individuals’ data. Others are looking into whether...