Some apps show a default action sheet in iOS 6 with sharing options.

Social Framework only has two classes, one for composing and one for request.

What I found though is about composing for a particular service with SLComposeViewController and before showing this I must query by hand if the service is available. And then I also have to create my own action sheet with own icons.

How do those apps show this default share options action sheet in iOS 6? Or are they using an open source framework?

The UIActivityViewController stated in the other answer makes this trivial. All you have to do is specify the text/image/URL that you want to share and present the activity view controller modally and iOS will automatically display all applicable sharing services. Examples:

Objective-C

- (void)shareText:(NSString *)text andImage:(UIImage *)image andUrl:(URL *)url
{
    NSMutableArray *sharingItems = [NSMutableArray new];

    if (text) {
        [sharingItems addObject:text];
    }
    if (image) {
        [sharingItems addObject:image];
    }
    if (url) {
        [sharingItems addObject:url];
    }

    UIActivityViewController *activityController = [[UIActivityViewController alloc] initWithActivityItems:sharingItems applicationActivities:nil];
    [self presentViewController:activityController animated:YES completion:nil];
}

Swift

func share(sharingText: String?, sharingImage: UIImage?, sharingURL: URL?) {
    let sharingItems:[AnyObject?] = [
                                        sharingText as AnyObject,
                                        sharingImage as AnyObject,
                                        sharingURL as AnyObject
                                    ] 
    let activityViewController = UIActivityViewController(activityItems: sharingItems.compactMap({$0}), applicationActivities: nil)
    if UIDevice.current.userInterfaceIdiom == .pad {
        activityViewController.popoverPresentationController?.sourceView = view
    }
    present(activityViewController, animated: true, completion: nil)
}