Important: This check should always be performed asynchronously. The majority of answers below are synchronous so be careful otherwise you'll freeze up your app.

Swift

1. Install via CocoaPods or Carthage: https://github.com/ashleymills/Reachability.swift

2. Test reachability via closures

   let reachability = Reachability()!
   
   reachability.whenReachable = { reachability in
       if reachability.connection == .wifi {
           print("Reachable via WiFi")
       } else {
           print("Reachable via Cellular")
       }
   }
   
   reachability.whenUnreachable = { _ in
       print("Not reachable")
   }
   
   do {
       try reachability.startNotifier()
   } catch {
       print("Unable to start notifier")
   }
   

Objective-C

1. Add SystemConfiguration framework to the project but don't worry about including it anywhere

2. Add Tony Million's version of Reachability.h and Reachability.m to the project (found here: https://github.com/tonymillion/Reachability)

3. Update the interface section

   #import "Reachability.h"
   
   // Add this to the interface in the .m file of your view controller
   @interface MyViewController ()
   {
       Reachability *internetReachableFoo;
   }
   @end
   

4. Then implement this method in the .m file of your view controller which you can call

   // Checks if we have an internet connection or not
   - (void)testInternetConnection
   {
       internetReachableFoo = [Reachability reachabilityWithHostname:@"www.google.com"];
   
       // Internet is reachable
       internetReachableFoo.reachableBlock = ^(Reachability*reach)
       {
           // Update the UI on the main thread
           dispatch_async(dispatch_get_main_queue(), ^{
               NSLog(@"Yayyy, we have the interwebs!");
           });
       };
   
       // Internet is not reachable
       internetReachableFoo.unreachableBlock = ^(Reachability*reach)
       {
           // Update the UI on the main thread
           dispatch_async(dispatch_get_main_queue(), ^{
               NSLog(@"Someone broke the internet :(");
           });
       };
   
       [internetReachableFoo startNotifier];
   }
   

Important Note: The Reachability class is one of the most used classes in projects so you might run into naming conflicts with other projects. If this happens, you'll have to rename one of the pairs of Reachability.h and Reachability.m files to something else to resolve the issue.

Note: The domain you use doesn't matter. It's just testing for a gateway to any domain.

In Xcode's Organiser, go to your device's Summary tab. Find your application in the list, and click the disclosure triangle. Under it, you should see an icon saying "Application Data". Click the down pointing arrow to download the data, and it'll prompt you for somewhere to save it.

In Xcode 5, listed under your device in Organizer, click on "Applications" and you can see "Data files in Sandbox" in the bottom half of the window.

In Xcode 6, go to Window -> Devices, select the device, select the app name under Installed Apps, click the settings gearbox, then select Download container.... In finder, double finger tap the package and select Show package contents.