What openings are the least sharp, the least tactical?

The King's Gambit Accepted is considered sharp and tactical. The King's Gambit Declined, less so.

When interested in a quiet, positional game, what openings would you recommend? With White, I would be happy to open with 1.e4 or 1.d4.

When looking for a quiet defense playing Black, what would you suggest for 1.e4 and 1.d4? Thanks ⛺

Against e4 Caro-Kann I guess. Against d4, try to play (or transpose) into Slav, Semi-Slav or QGD. Basically, most book moves (excepting gambits and practically all Sicilian lines) are quiet.

Maybe the Catalan.

As white, London system

As black, QGD against 1.d4, but against 1.e4 it's difficult to avoid some tactics - maybe the Center-Counter (with Qxd5, Qa5 and c6) is relatively quiet.

the english and the reti generally give quite quiet games, but you simply cannot and more to the point should not try to avoid tactics in chess.

 I think everyones been spot on with all thier suggestions. But I would like to mention two other King pawn games: 1) the Exchange French if you like watching glaciers move in real time. & 2) The Petroff with its series of equalizing exchanges not unlike being beat to death for 9 & a half hours with a nerf bat.

I'm going to say 3...Qd6 in the Scandinavian is more solid than 3...Qa5. The recent book The Scandinavian Move by Move is 400 pages and is about the 3...Qd6 ...c6 Scandinavian. It's a good book. He devotes a big chapter to 5...g6 for when black wants to play something different. It's caused me to switch to 3...Qd6 instead of the Caro Kann. The Caro Kann is so loaded in theory now and the pawn structures are all over the place depending on white's setup.

3...Qd6 Scandinavian along with the 2 Bf4 London (not the 2 Nf3 London) and 4...a6 Slav make for a very solid repertoire that often times have similiar development themes. Win with the London System is the only book that covers the 2 Bf4 London, and The Chebanenko Slav According to Bologan book is a must have for playing the 4..a6 Slav. When white doesn't play the 1 d4 2 c4 3 Nf3 4 Nc3 mainline setup, then instead of bothering with 4...a6 black can play it as a London reversed. 

Anything with "system" in the name... Igor Glek System (4 Knights or Vienna with g3,) Closed Sicilian with White, the "lion" system (which is an attempt to reach Philidor Defense positions with a different move order to avoid some bad lines... I don't know who well it actually works, but it mgiht be worth a try) with Black or similar play up a tempo with White: http://www.vanrekom.nl/thelion/indexgb.htm. The Hungarian Defense. I play the Czech Benoni which some people consider a "system" but I think there are a few problem lines if you try to play it like one and I think people have already given better suggestions against 1. d4.

I'll add to this the Petroff against 1.e4.

Because apparently trying to steamroll your opponent with your Kingside pawns isn't tactical.

Ok, Mainline_Novelty, which Sicilian is less tactical than the closed, the c3? (I don't know that one.) Also in general, no, I don't feel a Kingside pawn storm is a tactic. It is a positional stratagy... though I think I do get your point; you can't really avoid tactics at all stages in the game.

White's plans in the Closed Sicilian are most definitely agressive, and for me really, I find tactical and agressive to be very similar. Even the c3 Sicilian can be tactical/agressive though, especially in the 2...d5 IQP lines. It is in general hard for White to avoid tactics against the Sicilian, which is why a positional/quiet repertoire should probably be based around 1.d4 (although Be2 Open Sicilians may fit the bill)

I was thinking more on the lines of tactical relates to open positions, positional relates to closed positions. It's absolutely possible to be agressive in a way that your plan is deep, you grab space slowly, keep the position closed, and are unlikely to be undone by a single sharp overlooked combination. Admittedly most of the openings I suggested were passive, they do tend to go together, but I do think it's possible to have one without the other, and I stand by my suggestion of the closed Sicilian, though you're probably right 1. d4 is a better suggestion here. I just don't play 1. d4. (I also don't think the KG declined is necessarily less tactical or sharp, btw, I mean 2... Bc5 is a tactical defense of a pawn already and it's easy to go wrong by a single move for either color in those lines too. I would say that variation is certainly less agressive than accepted though.)

I suppose the best answer would be the opening with the least variations/responses, I would say the closed sicilian. It is quite straight forward in comparison the open sicilian or say the tactics behind the Kings Indian attack or Queens gambit declined.