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Time signatures are available in the main palette sidebar. You can drag and drop the time signatures onto the score (see Palette for general information on working with palettes in MuseScore).
Create your own Time Signature
If a time signature is needed that is not found in the corresponding workspace palette, open the Master Palette directly to the Time Signature section (ShiftT) to create your own. You can edit the numerator and the denominator in the Create Time Signature Panel by pressing the Add button. Once added, just drag and drop the time signature to the score from the window where you created it. Be aware that it will not appear in the workspace palette. If you want to add it in the palette, read Custom Palette.
Change default beaming
To adjust beams from the automatic offering, you can click the notes you want to modify.
For example: by default 5/8 is beamed 3+2. You can click the third and then fourth note to beam it 2+3. Don't forget to do it for the other two subdivisions. To edit the third one drag an icon from 
Delete a time signature
Time signatures you don't need (any longer) can be removed in the above dialog box via right-click → "Delete Content".
Edit to additive meters
In most cases, you'll only need to edit the first of the upper numbers. The additional upper numbers are for additive meters, which contain multiple upper numbers separated by a plus sign.
Different duration from time signature: Pickup measures (Anacrusis) and Cadenzas
There are occasions when the actual duration of a measure is different from the duration specified by the time signature. Pickup measures and Cadenzas are a common example. To change the actual duration of a measure without displaying a different time signature, see Measure operations: Properties, Measure duration.
Local Time Signatures
Time signatures can be different for different staves. An example here is Bach's 26. Goldberg Variation:
MuseScore has the concept of a global time signature and an actual (local) time signature. To change the global time signature drag and drop a palette object to a staff. The global time signature is used to count beats (as shown in the status line) and is the reference for tempo markings. The global time signature is the same for all staves and normally identical to the actual time signature.
The actual time signature is set in the time signature property dialog and can deviate from the global time signature for every staff (left hand 18/16 in the example).
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The text of the time signature can be set independent of the actual values.
A local time signature is set by dropping a time signature symbol while holding the Ctrl key. The local time signature is set only for one staff. A global time signature is replicated for all staves.
Time signature changes and Other Objects
Multimeasure breaks are interrupted when a time signature change occurs.
Also, a section break will prevent a courtesy time signature being shown at the end of the previous measure.