Where do the Patrons reside? Well, that's both easy and hard to say.
Lady Oshtur dances across the night sky along with her children, those of the Zodiacal March. You can see the path they stamped out in silver in the heavens. Rarely she dreams of past loves or her lost sister, and then you can see Her veils of green, red, and white in the sky to the north.
Lady Ishtar has a mansion of massive timbers in an ancient forest. There are all kinds of fruit trees, flowers, crops, and bees buzzing about. The lands surrounding are filled with the sounds of running water, streams, rivers, majestic waterfalls. No one is ever thirsty there, I can tell you. Her hall is in Anakarum, where the giants live. But, it is a peaceful place, and there is no fear.
Lord Brisingr travels constantly, often between Ladies Ishtar and Kirigal's halls. While He can rest easy in Lady Ishtar's repose, it is a fitful slumber for him under Lady Kirigal's slate roof in Acheron. The moans of Lord Nergal from his unhealing wound make any comfort found in Lady Kirigal's home spare indeed.
Lady Kirigal, as mentioned, has her manse in Acheron with her husband Lord Nergal. She is seldom home being of restless spirit. We see her all too often riding her storms from the West, East and South. Still she returns home to tend to Lord Nergal's Wound, allowing Him to find some rest while She is there.
Lord Marduk has his manse in Asphodel. There He learns and records the methods of all arts, crafts, and techniques for the betterment of mortals such as us. Statecraft, metalwork, fishing, reading the wind. All this he catalogs while keeping his spear close by should the Walls of Dis crumble.
Lord Bahamut gazes from his tower on the Horizon's Rim, keeping watch against the incursion of demonic might. The Seven-Eyed Sage sees all that crawls, swims, and flies by light of day or night. Demon's Guard can be found on the horizon, East of the Sun and West of the Moon.
Lastly, Lord Asmoder sits on His throne in Carceri overseeing the prison of the Iron Sun. Far, far away from us, not that we would ever want to go there. That's where the infernals reside, sometimes tempting mortals to join them in their eternal watch.
There are the others of course. We usually talk about the Seven Patrons, although there are really twelve. The Septeon are those that care most for us mortals, and who help us in need. There's also Lady Mharg whose manse is far below us. She's the Patron of the dvergar - Lord Neander beseeched Her for succor long ago and She actually agreed. Lady Sekhalah occupies Lady Pelagra's Palace of Coral and Shell since it stood vacant for ages. Now She owns the deeps and the secrets they hold. I mentioned where Lord Nergal's rests. Lady Nagamat never rests in the same place twice, eternally exploring the Spindle. Lord Zalgo's home is non-existent, being in a place between realms rather than in one. Their truest desire is to be left alone, utterly alone. Disturb It Not.