It is hard, very hard. It's probably not what you want to hear but I can't actually think of any practical advice 
The exhaustion from someone working physical, demanding, probably stressful shifts and sleeping at really odd times throughout the week is pretty overwhelming so yes, while sometimes it seems unfair to outsiders, they probably do need that lie in. And yes, weekends rarely seem to coincide with them having a few days off so that you can have a lie in too.
How old is your DC? Are you working too?
The only benefits we have at the moment with DP doing emergency services shifts is that his seem a bit more fixed, so I can work f/t too and he probably does more daytime childcare than I do. I agree with a PP, DP occasionally puts his foot down and says that he has sole charge of DD and so he can't do an extra shift/work overtime.
Social life - yep, it's babysitters all the way. Or going on your own with your DC (which I hate too. I went to a wedding last summer without DP and with a toddler, it should have been fantastic, it was hell on earth.)
My support is often my parents, too. They aren't near enough to help out on a day to day basis but they are near enough to visit, so every few months I just go there at the weekend with DD and have a chance to get some proper sleep. It works for us because it's a intensive but short period of them toddler wrangling, rather than me calling on them for an hour here, an hour there.
But for all that, I think the big thing when you have a partner who does this sort of job is that you actually just have to get on board with it. It probably isn't going to change. If they are a police officer, paramedic, A&E doctor -it's shiftwork for the foreseeable future. Nothing is going to change. I know that can seem so insanely depressing, but IMO you have to just face up to it. Do you love them? Do you want to be a family together? Yes? Then be honest, talk to each other about the problems you are having and where you can help each other, but the big thing, for me, is that you can't keep arguing about the job and the hours.
Though if he's getting a full 8 hours every night, then poke him in the ribs after 7 hours sometimes and tell him to get up and give you at least one hour's peace 