I am a Nigerian, holding a valid schengen visa issued by the French embassy, and a multiple entry visa for the United States. Will I be allowed to enter Tbilisi in Georgia?
According to https://matsne.gov.ge/en/document/view/2867377:
List of Countries, Whose Visa and/or Residence Permit Holders May Enter Georgia without a Visa for an Appropriate Period and under Appropriate Conditions
United States of America
Aliens holding a visa and/or a residence permit of any country listed in the annex may enter and stay in Georgia without visa for 90 calendar days in any 180-day period.
For the purpose of entering and staying in Georgia without visa, a visa and/or a residence permit of a respective country shall be valid on the day of entry into Georgia (crossing the border), which must be evidenced by a travel and/or other appropriate document.
You can also fill out a few details about yourself (intent of travel, citizenship, ...) at https://geoconsul.gov.ge/en/visaInformation and it will tell you whether you need a visa.
In general, you should be good to go, assuming your US-visa is still valid.
As stated in Timatic, the database used by airlines:
Visa required, except for passengers with a valid visa issued by an EEA Member State, Australia, Bahrain, Bermuda, Canada, Cayman Isl., Falkland Isl. (Malvinas), Gibraltar, Israel, Japan, Korea (Rep.), Kuwait, New Zealand, Oman, Qatar, Saudi Arabia, Switzerland, Turks and Caicos Isl., USA, United Arab Emirates or Virgin Isl. (British) for a maximum stay of 90 days.
So yes, you can enter Georgia on the basis of your French or US visa.
As a Nigerian citizen, you will typically need a visa to enter Tbilisi, Georgia. Nigeria is not on the list of countries whose citizens can enter Georgia visa free. The complete list can be found here
However bearers of valid French visas or valid USA visas can enter Georgia without a visa per the Legislative Herald of Georgia.
You can also find a simplified map for visa requirements to enter Georgia on Wikepedia.
Holders of visas or residence permits of EU/EFTA/GCC countries, overseas territories of EU countries (except Anguilla, Montserrat, Pitcairn, Saint Helena, Ascension and Tristan da Cunha), Australia, Canada, Israel, Japan, New Zealand, South Korea or the United States do not require a visa for max 90 days in a 180-day period. The visa/residence permit must be valid on arrival to Georgia.
1OP didn't say anthing about a UK visa. Also, I'm not sure whether your information is up-to-date. Mar 6, 2017 at 20:53
@martin.koeberl Thanks for the pointer. Noted and updated. Mar 6, 2017 at 21:05