Cups, 1870s

The oldest Cup competition, the Football Association Challenge Cup, was introduced in the 1871-72 season. The Scottish Football Association Challenge Cup followed in 1873–74.

... the spread of the game through the 1870s and 1880s was largely bound up with the principle of the Cup competition. That was where the real enthusiasm lay.
Geoffrey Green in  Association Football (1960)

Here are ten other cups that have been played  for since the 1870s:

The Birmingham Senior Cup 1875/76

1875-76 Tipton 1–0 Aston Villa
Tipton were not connected to the present Tipton Town.

Wednesbury Old Athletic
1876-77 Wednesbury Old Athletic 3–2 Stafford Road

1877-78 Shrewsbury FC* 2–1 Wednesbury Strollers

1878-79 Wednesbury Old Athletic 3–2 Stafford Road

1879-80 Aston Villa 3–1 Saltley College

* Some sources give Shrewsbury Town , a club that  was not formed until 1888. The Birmingham County Football Association were unable to offer any information on the identity of the winners. 

soccerdatauk  members soon came up with some great information about this  earlier Shrewsbury club.

The East of Scotland Shield 1875 (The Edinburgh F.A. Cup)

1875-76 Third Edinburgh Rifle Volunteers 6-0 Thistle
1876-77 Thistle walk over Third Edinburgh Rifle Volunteers (failed to show up for the final)

 Heart of Midlothian
1877-78 Heart of Midlothian 3–2 Hibernian (4th replay)
1878-79 Hibernian 2–0 Heart of Midlothian (replay)
1879-80 Hibernian 5-0 Dunfermline (replay- Dunfermline lodged a protest after the first match, which Hibernian had won 6-3)

The Sheffield and Hallamshire Senior Cup 1876/77

1876-77 The Wednesday 4–3 Heeley

The Wednesday
1877-78 The Wednesday 2–0 Attercliffe
1878-79 Thursday Wanderers 3–1 Heeley
1879-80 Staveley 3–1 Heeley

 The Welsh Cup  1877-78 

1877-78 Wrexham 1-0 Druids
1878-79 Newtown White Star 1-0 Wrexham 
1879-80 Druids 2-1 Ruthin

The Shropshire Senior Cup 1877-78

1877-78 Shrewsbury F.C 1-0 Wellington
1878-79 Newport
1879-80 Shrewsbury Engineers

Thanks to Zoe Griffiths of the Shropshire Football Association for providing me with the winners -
Shrewsbury F.C also won The Birmingham Senior Cup in 1877-78 (thanks to soccerdatauk members for their help with this).
I was unable to find the results of the 1879 or 1880 finals, but I did find the following intrigue in the local press of the time:
In 1880 The Shropshire Football Association took legal action against Mr A.T Ward, captain of Newport FC. Apparently when Newport won the cup there was no trophy, but the Shropshire FA had £23, which they agreed to give to Mr Ward on the understanding that he would raise the remaining money necessary to buy a trophy. When  Shrewsbury Engineers won the competition the following year Mr Ward refused to give the cup to the Association until the balance had been paid.The Association wanted either £40 plus £5 expenses or the cup itself. The court ordered Mr Ward to hand over the cup.

 The North Staffordshire Challenge Cup 1877-78

1877-78 Stoke 1–0 Talke Rangers 
1878-79 Stoke 2–1 Cobridge
1879-80 Wednesbury Old Athletic 2-1 Aston Villa

The Berks & Bucks Senior Invitation Cup 1878/79

1878-79  Reading 1-0 Marlow (2-0 in some sources)
1879-80 Swifts 1-1 Old Philberdians (replay drawn: trophy shared)

The Renfrewshire Cup 1879/80

1879-80 Thornliebank 2-0 Arthurlie 

The Cheshire County Football Association Challenge Cup 1879/80

1879-80 Northwich Victoria 2-0 Hartford St Johns

The Lancashire County Football Association Cup 1879/80

1879-80 Darwen 3-0 Blackburn Rovers