From the docs:

jq '.[] | select(.id == "second")' 

Input [{"id": "first", "val": 1}, {"id": "second", "val": 2}]

Output {"id": "second", "val": 2}

I think you can do something like this:

jq '.theList[] | select(.id == 2 or .id == 4)' array.json

A slightly different approach from what rowboat took:

jq 'map(select(any(contains("a"))))' file

The innermost test tests each of the strings for the substring a. If any string in a sub-array contains a, that sub-array is selected. The map applies the selection to all sub-arrays in the top-level array.

The result is a top-level array with sub-arrays that contain at least one element that contains the substring a:

[
  [
    "a",
    "b",
    "c"
  ],
  [
    "abc",
    "e",
    "f"
  ]
]

Very close! In your select expression, you have to use a pipe (|) before contains.

This filter produces the expected output.

. - map(select(.Names[] | contains ("data"))) | .[] .Id

The jq Cookbook has an example of the syntax.

Filter objects based on the contents of a key

E.g., I only want objects whose genre key contains "house".

$ json='[{"genre":"deep house"}, {"genre": "progressive house"}, {"genre": "dubstep"}]'
$ echo "$json" | jq -c '.[] | select(.genre | contains("house"))'
{"genre":"deep house"}
{"genre":"progressive house"}

Colin D asks how to preserve the JSON structure of the array, so that the final output is a single JSON array rather than a stream of JSON objects.

The simplest way is to wrap the whole expression in an array constructor:

$ echo "$json" | jq -c '[ .[] | select( .genre | contains("house")) ]'
[{"genre":"deep house"},{"genre":"progressive house"}]

You can also use the map function:

$ echo "$json" | jq -c 'map(select(.genre | contains("house")))'
[{"genre":"deep house"},{"genre":"progressive house"}]

map unpacks the input array, applies the filter to every element, and creates a new array. In other words, map(f) is equivalent to [.[]|f].

The -r option should not be used if the output will be fed into another JSON parser; only the last jq in the pipeline should use it – otherwise you risk feeding non-JSON data into the second jq instance.

Your last example with .[0].t seems correct and works with jq 1.6.

$ cat json | jq '.tx' | jq -r '.[0].t'
t for a

But this can be simplified to a single jq invocation, first incrementally to:

$ cat json | jq -r '.tx | .[0].t'
t for a

and then finally to:

$ cat json | jq -r '.tx[0].t'
t for a

If "rebuilding" your structure is an option you could do something like this

jq '.Vpcs[]|select(.OwnerId!="abc")|{Vpcs: [.]}'