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Whenever I book a flight through any travel agency/website; they always make a data entry of my name as "BADWAL/SHUMSMR". However, this time I decided to book with Expedia India; and they just inputted my name as "BADWAL/SHUMS" without appending the MR post-fix.

Due to this, when I check my ticket details on the British Airways website, my name shows up as Ms. Shu Badwal; this is probably because my name ends with MS. Would this cause me to have any problems while checking in?

This will almost certainly not cause you any issues, and if it does they are problems with will be able to be fixed very easily by the BA staff.

There is no requirement to append "MR", "MRS", "MS" or similar to the name on a ticket, although it still does occur sometimes. Both the versions with and without the title are completely valid.

In this case the BA website is seemingly attempting to parse your name, and simply displaying it incorrectly. As long as the ticket itself has a valid version of your name then at most this might cause a few second delay when checking in - although I'd suspect that even that is extremely unlikely.

Although it's less rare now, it used to be that the title was put before the firstname, and frequently only an initial was used for the first name. My first name starts with "S" and I am male, so my tickets often became SURNAME/MRS and more than once I was addressed as "Misses Surname", however it never caused any issues other than some very minor confusion.

I think depending on the departure locale, this can definitely cause you problems checking in. For instance, if you are departing from a busy airport in India such as Mumbai, you may not gain entry to the airport if the guard doesn't think your name matches between the check-in list and your passport.

Whereas, if you are departing from a British Airways hub such as London Heathrow, there will be plenty of British Airways staff who are directly available to assist you and listen to your explanation.

So, depending on your situation it can be very beneficial to spend some time communicating with an Expedia or British Airways representative to change your name record. I would advise first to try British Airways.

As Doc described in his excellent answer,

This will almost certainly not cause you any issues

But if you are still nervous about this, you can call the travel agent and ask them to add the MR suffix to the reservation as a name update.

Most airlines have a distinction between name changes (where you rebook the ticket to a different passenger) and name updates (fixing small spelling mistakes or adding a second name, but the passenger is still the same person). Usually name changes incur a change fee, but name updates are free.

This is true for most airlines that are not low-cost, and indeed the BA website tells us that if

the name of a person travelling is spelt wrong or doesn't match their name on their passport: Most spelling mistakes in the name of a traveller can be easily corrected over the phone, when all of the flights on their ticket are operated by British Airways. We don't charge for this type of change, however if the taxes, fees, carrier imposed charges or fuel surcharges on the ticket have changed since the booking was made we'll need to take any additional money due for them at this point.

(Note the catch at the end -- it's up to you if you want to take this risk).

However, note that

If you booked through a travel agent or on a travel website, please contact them directly. We can't change tickets they have issued.

For completeness, as this is not applicable in your case, but for future reference: BA has a 24 hour 'cool off' period in which you can cancel the booking and get a full refund regardless of the reason. So if they get your name completely wrong, immediately contact them.

(Source of the quotes: BA Website)