Whenever I'm adding a new contact (iOS 9.3.4, German) and tap "add phone number", the default phone number type is "Home Fax" which is probably the least useful default imaginable. Is it possible to set the default to "Mobile"?
The weird default values continue when I tap the entry to change the type – the list that pops up is ordered "Home Fax, Radio (??), Home Phone, Mobile Phone, …" – I would prefer a different order like "Mobile, Home, Work, Work Fax, Home Fax, …" or something similar, but I can't find a way to change that order.
It appears to be related to the service you are using.
Open Settings, find Contacts, and see what it's set to for Default Account. Apparently the service provides the list of account types.
Mine is set to Exchange, so when I create a new contact, and add a phone, it defaults to Home Fax.
If I change the Default Account to iCloud, it defaults the phone type to Phone.