First of all, you need to format your code for reddit (four spaces at the start of each line) or it's impossible to read:

current_users = ['May', 'April', 'Wu', 'Su', 'Lulu']
new_users = ['may', 'mike', 'jones', 'chow', 'paul']
for new_user in new_users:
    if new_user in current_users:
        print('username '+new_user+' is not available.')
    else:
        print('welcome')

There's several ways to do this. One is to create a new list that has all lower case in it:

current_users_lower = [x.lower() for x in new_users]
for new_user in new_users:
    if new_user.lower() in current_users_lower:
        # do things

You should convert your current_users into a lowercase set and then do blazignly fast comparisons for each of your new users, just lowercased, e.g.:

current_users = ["John", "Admin", "Jack", "Ana", "Natalie"]
new_users = ["Pablo", "Donald", "Calvin", "Natalie", "Emma"]

current_users_lookup = {user.lower() for user in current_users}
for user in new_users:
    if user.lower() in current_users_lookup:
        print("Username {} unavailable.".format(user))
    else:
        print("Username {} available.".format(user))

Which would get you:

Username Pablo available.
Username Donald available.
Username Calvin available.
Username Natalie unavailable.
Username Emma available.

Actually you can make this in just one list comprehansion:

list_A = ['Sasi', 'Babu', 'kuttappan', 'mathayi']
list_B = ['Raman', 'Kesavan', 'sasi', 'unni', 'Kuttappan', 'SaSi']

duplicated = [b for b in list_B if b.lower() in (a.lower() for a in list_A)]
print(duplicated)

This way it returns the original values while comparing the lowercased. Using sets will return the lowercased values and will delete all duplicated values in list_B.

fruit.lower() in the for loop won't work as the error message implies, you can't assign to a function call..

What you could do is create an auxiliary structure (set here) that holds the lowercase items of the existing fruits in fruits, and, append to fruits if a fruit.lower() in fruit_add isn't in the t set (containing the lowercase fruits from fruits):

t = {i.lower() for i in fruits}
for fruit in fruits_add:
    if fruit.lower() not in t:
        fruits.append(fruit)

With fruits now being:

print(fruits)
['Apple', 'banana', 'Kiwi', 'melon', 'strawberry']

Assuming ASCII strings:

string1 = 'Hello'
string2 = 'hello'

if string1.lower() == string2.lower():
    print("The strings are the same (case insensitive)")
else:
    print("The strings are NOT the same (case insensitive)")

As of Python 3.3, casefold() is a better alternative:

string1 = 'Hello'
string2 = 'hello'

if string1.casefold() == string2.casefold():
    print("The strings are the same (case insensitive)")
else:
    print("The strings are NOT the same (case insensitive)")

If you want a more comprehensive solution that handles more complex unicode comparisons, see other answers.

The example you've given seems to work, but I'm going to assume that current_ids may have a mixture of upper/lower case.

What you could do is convert every string in current_ids to lower case by using a list comprehension:

current_ids = [i.lower() for i in current_ids]

This creates a new list of every word in current_ids all lower case.

Then, you can compare as you are doing now (if new.lower() in current_ids)

You can convert each new username to lower case and compare it to a lower case version of the user list (created using a list comprehension).

lower_case_users = [user.lower() for user in current_users]
if new_user.lower() in lower_case_users:
    # Name clash

You may want to store your usernames in lowercase when they are first created to avoid creating a lower case version from the user list.

That is because your else condition executes before the last if condition. So here I have change the order.

current_users = ['nedboy78', 'codingking678', 'johnnykapahala','jam95','python65','ANDY657']
new_users = ['hamlet56', 'python65', 'jam95','todds4','andy657']

current_users_convert = [current_user.lower() for current_user in current_users]
for new_user in new_users:
    if new_user in current_users:
        print(f'sorry username {new_user} not available pick a new username')

    elif new_user in current_users_convert:
        print(f"CANT USE {new_user} AS A USERNAME")

    else:
        print(f'username {new_user} is available')

Output:

username hamlet56 is available
sorry username python65 not available pick a new username
sorry username jam95 not available pick a new username
username todds4 is available
CANT USE andy657 AS A USERNAME

You should create a set that contains all of the lowercased names. Then, to check whether a name can be used, look up the lowercased version of the name in the set:

names_in_use = {name.lower() for name in current_users}

for new_user in new_users:
    if new_user.lower() in names_in_use:
        print (f'The username {new_user} is already in use. You need to enter a new username')
    else:
        print (f'The username {new_user} is avaliable')

This outputs:

The username Janice is avaliable
The username Ben is avaliable
The username Phoebe is avaliable
The username JOEY is already in use. You need to enter a new username

username = 'MICHAEL89'
if username.upper() in (name.upper() for name in USERNAMES):
    ...

Alternatively:

if username.upper() in map(str.upper, USERNAMES):
    ...

Or, yes, you can make a custom method.