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Disable the airplane mode
Maybe your device doesn't ring because of the Airplane mode. If you don't see the signal strength, nor the operator's name in the upper-left corner of the screen, but you see a small airplane icon, the airplane mode is enabled.
If liquid or dust found their way to the device's headphone port, they will be deceiving your device that headphones are plugged in it, which would explain why it doesn't ring. If you see a headphones icon near the battery icon, you are definitely facing this issue, popularly known as "stuck into headphone mode".
In order to clean the device's headphone port you will need a cotton bud and an isopropyl alcohol (90-100%), due to its evaporating property. Put a small amount of the alcohol on the bud and then place and rotate the bud into the headphone port, gently.
Check the video below to see how to clean the headphone port:
Maybe your iOS device doesn't ring when you receive calls from an app (e.g. Viber or Skype) because the notifications for that app are disabled in the Settings. In order to enable the notifications, follow the steps below:
Go to Settings and tap on Notifications
Tap on the app which doesn't ring and move all the sliders to the right.
Your device might not ring because of misconfigured settings. Resetting the app preferences will restore the default settings of the apps. Follow the steps below in order to reset the apps' preferences.
Go to Settings and select Apps.
Tap the three-dot icon and select Reset app preferences.
-- Note: The screenshots are taken using a Sony Xperia M2, running on Android 5.1.1 Lollipop. The instructions mentioned here might slightly vary from one manufacturer to another.
Repair the mobile device using the software that comes with it
Every mobile device has its "PC Companion" software, whose purpose is repairing the smartphone's operating system, making backup of personal data, etc. For example, iPhones have iTunes, Sony mobile devices have Sony PC companion, etc.
Connect your device to the computer via a USB cable, (install and) run the software, follow the instructions by the software and wait for it to repair your device.
Note: If you don't have the software installed in your computer, you can download it from the manufacturer's website. Also, make sure that you make a backup of all your personal data, such as photos, videos, sounds and other documents.
If your iPhone doesn't ring, but you receive notifications in your voicemail that somebody called you, it's possible that you have enabled the Do Not Disturb option. If you see a half-moon icon next to the battery icon, this option is enabled.
You can disable this option via the Action Center or via Settings:
Via the Action Center:
Swipe up from the lower part of the screen and tap the half-moon icon.
Go to Settings -> Tap Do Not Disturb -> Move the switch next to Manual to the left.
If you tried all the other solutions, but none of them worked, reset your device to its factory settings. Before you reset it, make sure to make a back-up of all your personal data, such as pictures, videos, sounds, etc., because they will be deleted from your device.
If you tried everything else, but you were unable to fix the app which doesn't ring when a notification arrives consider uninstalling the app, via Google Play (Android devices), App store (iPhone) or Windows store (Windows phone OS).
Maybe the speaker itself has malfunctioned and is unable to ring. If you tried all the other solutions and none of them worked and if you feel comfortable with disassembling your device on your own replace the speaker.
In order to perform this procedure, you will need a new speaker, which you can purchase in authorised repair shops and disassembly kit, which includes screwdrivers and other tools.
If you are not sure how to disassemble a device, type your device's model number followed by "disassembly" or "speaker replacement" in order to find a tutorial.
For instance, this is one of the tutorials you get if you type Samsung Galaxy S5 speaker replacement:
Note: Be very careful when you disassemble your device as it is very fragile.
If you have turned down the device's volume, your device won't ring. Press the volume (+) up button to turn up the volume and check if your device starts ringing, by calling it from another device, sending an SMS, etc.
If your device doesn't ring when you receive notifications from an app, it is possible that its notification sounds are disabled. In order to enable them, you should go in the app's notification settings.
It is possible that the ringtone you set to notify you when someone is calling you or when you receive notifications of any kind is deleted or moved to a different location. Try setting another ringtone in the device's settings.
If you use a BlackBerry Bold phone, remove the back cover of the phone. Then press the very-small plastic button (white or light grey color), which is located above the flashlight, with finger nail or a little pair of scissors.
It is very important to update the device's operating system on time as the updates contain fixes to various bugs. If you don't receive a notification about the update, check if one is available manually.
The call diverting feature enables you to divert all the incoming calls from one phone number to another. If you have enabled this feature, you won't be able to receive calls; therefore, your device won't ring.
Go to the device's Call Settings and disable call diverting. If you are not sure how, ask the carrier (operator) you are subscribed to for instructions.
If the mute function is enabled in the Motions and Gestures menu, all the incoming calls will be muted if you cover the phone's front part (the screen) with your hand or with any flat surface. To prevent this from happening in future, disable the mute option in the Motions and Gestures menu.