Which Branch Should I Join as a Civil Engineering graduate (Army, Navy or AF)?

It depends on what you see as your ultimate future. That's when you're done with the MIL stuff. Take a look at the varied progression plans for engineers. If your plan is to be a registered PE, then you want a program that will support it. For the Navy, it's pretty simple. You WILL get your Seabee Warfare Quals done. You WILL go to Grad School, You WILL get your Contracting Warrants. You WILL get your PE ticket. That's if you want to make O-5. Risky going for O-4 as the people who don't have it usually are in the third crunch at the promotion board. The Navy plan is pretty specific and you need to be shrewd in making sure you hit all the pedigree buttons. I'm sure the AF and AR have specific preferred paths. The problem with the AF is much of their construction contracting is done by AR and Navy, so there's experience you'd find hard to get. AR experience that translates best to the outside is COE work. They do facilities, real estate, waterways, dams, etc. Those are different billets vs. demolition, sapper, revetments, etc. which have limited commercial application.

BTW, registration for PE is trending to require more schooling than a BS. Your Grad work needs to be in an engineering discipline that you want to get your ticket in. It was a terrific journey for me in doing real engineering on all the continents (yes, Antarctica too). To do what you're contemplating, you'll have to be evermore moving forward, lest you never get there and become disillusioned. But the journey will be worthwhile. Remember the old Chinese Proverb: "May you live in interesting times."

If you want to actually do engineering the federal service is not for you. There is very, very little actual engineering going on by federal staff, military or civilian. It is much easier and better to have the concept drawings done by in staff personnel, but the actual engineering is done by a contract, awarded to an A&E Firm that specializes in the type of building to be designed. As such, if you really want to design and do engineering commissioning into the reserves or National Guard is the best place for you, with a full time job at a private company.
If you would rather be active duty, I would highly recommend looking at the Navy, you'll get the best duty locations and never need to serve on a ship. Sure beats Minot North Dakota or El Paso Texas.
My background is an Army Engineering Officer who works his civilian job as a Federal Construction Contracting Officer.

Good luck.

PFC (Join to see), I can't comment on the Engineer side of the house and there are some great responses on here...some bad ones too. There is NO guarantee that you will get to be an Engineer if you join. Not in the Army at least. That's if you go through a program like ROTC. Your branch is determined by a board that meets and determines what you will do. Having a Civil Engineering degree would help, but it's no guarantee. You could end up as a Field Artillery Officer. If you can find a direct commission program, that's the only way to guarantee...but those are few and far between. Engineers in the military don't do what you are talking about, even in USACE.

https://www.gocoastguard.com/active-duty-careers/officer-opportunities/programs/direct-commission-engineer-dce

Look at this....

Army. Thank you for your future service.

https://www.navy.com/careers/engineering-applied-science/construction.html

I'm resending this, going commissioner this way could ne perfect for you...

Army fool

I suggestion if you want to do engineering try getting a position through usajobs with the corp of engineers and not through joining the Army. You are more likely to get into the position you are looking for than if you join and hope to be appointed to the engineering core.

http://www.mspe.org

Here's their link....

I'm not saying Corps of Engrs is a bad way to go, I'm merely offering you alternatives, that you might not know about, NOAA is pure science, plus, they want masters only minimum now, and have an excellent flight program. And by flight, I mean back seat also, not just pilot, I'm speaking of USAF nav.UNT, and Navy NFO, USCG also has an excellent flight program. At your level, you'd have a decnet chance to get into AFIT or NPS for USAF or Navy, their grad schools, look them up, I'll try to send you the links, as well, or you could also do a bioscience PhD at USUHS and branch out that way, USPHS has serious use for engineers, plus, USPHS can also detail you back to one of the other services for you to be an engineer with them as well, I'd read of a USPHS engineer who'd gotten an award serving with Army. Look at all services, all, not solely one, trust me, I kmow exactly what I'm talking about, you'd have a good chance at flight test engineer with the USAF USN, UK, or French test political schools, even if you didn't go flightz or you could also try for flight test navigators or submit for NASA, though you'd likely need your PhD if you weren't flight qualified. Warrant might not ne a bad way to go also, there are numerous options for that, USCH has an R&D center in Connecticut, that'd ne perfect for you, give me more, and I'll try to help you pick from various directions, OK? And get your EIT and PE done, either that or land surveyor (LS), or registered architect (RA) possibly, join the NSPE, I'll send you the link, OK?