The answer is Australia.
Though it seems odd at first, there's very good rationale for answering this way.
Canada's passenger rail network is nearly nonexistent outside of the Windsor to Quebec corridor with only token of VIA "lifeline" services which, in reality, function more as tourist land cruises (though Vancouver does have limited commuter service linking the city to some of its eastern suburbs) providing passenger rail service. The sole bright spot is the Golden Horseshoe, where GO Transit runs a very comprehensive network of integrated rail and bus services.
On the surface, the United States performs best. Train service along the Northeast Corridor from Boston to Washington and along some key corridors in California is quite frequent. There are other corridors with five or more trains a day scattered about, including Seattle to Portland, Philadelphia to Harrisburg, and Chicago to Milwaukee. A few select places also have commuter rail networks which link most major destinations on a regular basis (northern Indiana, Connecticut, and eastern Massachusetts). The reality is, however, quite different. Large gaps even in the best-served areas exist, such as the lack of train service westwards from New York City to the Lehigh Valley and Scranton. Long-distance services, while not as tourist-heavy as those in Canada, still carry more leisure traffic than one would expect for a "lifeline" service and often serve non-coastal cities at off hours (for instance, all trains which stop in Cleveland serve it during the dead of night).
This leaves us with Australia. Why is Australia the best of the three? It has to do with the concentration of population, regional connectivity, and bus/rail integration. Compared to Canada and the United States, the Australian population is highly concentrated in a few select cities. Excluding Queensland and Tasmania, over half of the population of each Australian state (as well as the Northern Territory) lives in a capital city. The five major capital cities have extensive and frequent mainline rail networks. Sydney, Melbourne, and Brisbane have extensive regional rail networks which link each of them to nearby urban centers on a regular basis. Victoria in particular blows any comparable area in North America (with the possible exception of southern Ontario) out of the water, with 3 trains an hour between Melbourne and Geelong and hourly service to Ballarat, Bendigo, and the Latrobe Valley. New South Wales, Victoria, and Western Australia also have large intercity networks which connect to coach services and therefore create a network where long-distance bus services compliment train services instead of competing as is the case in Canada and the United States (assuming that such services even exist at all due to the decline of coach travel in the United States and Canada). Though these services aren't particularly frequent, they do operate on a regular basis and provide reasonably comprehensive coverage at the state level.
Of course, Australia's network isn't without problems. Western Australia's integrated bus and train network is limited to the southwest. The Northern Territory has only the tourist-targeted Ghan linking its two main population centers. South Australia has no non-tourist trains operating beyond the Adelaide commuter belt and Tasmania has no passenger train service at all. Service could be a bit more frequent along the Queensland coast and there is no regular train service between Brisbane and Toowoomba. Nevertheless, on a pound-for-pound basis, the network is much more comprehensive than those in Canada and the United States in terms of serving people and communities.
