Emulation Platforms (note - the Spectrum has a slow processor by modern standards, you can easily run Spectrum emulators on fairly old machines, eg 15 year old PC, 10 year old phone)

Browser

You're here, so you have a browser. Browser emulators are easy - no install, use on any device with an internet connection. On the downside, most are not completely accurate, and have few options for emulating extra hardware and none for debugging code.

There are a couple used by Spectrum Computing - JSpeccy and QAOP because you can instruct them to open software direct from the Spectrum Computing archive. For instance search this site for Jumping Jack, click for more details, and under Main File downloads you have the option to run the game from one of these emulators. 

JJ Options

Click on QAOP and the game is loaded before your eyes into a virtual Spectrum. Press ALT to go left, Space to go right and Shift to jump

Other browser emulators of note -

NameComments
RAZERelatively new and probably the most accurate browser emulator around. It has a virtual keyboard option that can help if your browser is on a phone.
internet archiveThere are many websites with retro games playable online. This one is interesting as it is a US non profit organisation hosting a lot of media including many Spectrum games, even Ultimate and Codemasters games that are not available elsewhere. It uses JSMame - Javascript Mame to emulate retro machines in the browser
ASM80Z80 development system online - you can write machine code and test it online.

Windows / Linux / Mac 

Emulators started appearing for DOS and Windows in the mid 90's. In the noughties they came to Mac and Linux also. There are more emulators for Windows than you can shake a joytick at, but also many that haven't been updated for 15 years or more.
A some of the more popular / interesting ones can be found in this Emulator Comparison Spreadsheet. There's a more select list below of a few good options More info on which ones are suitable for debugging machine code are on the Assembly Tools page

NameCommentsRuns on
SpinPopular, easy to use, good debugging options and built in assembler, but not updated since 2006 so may not run new games that use multi-colour effectsWindows
Fuse Popular, supports loads of extra hardware. Originally developed for Unix, but the emulator core is now used by many other emulators, such as RetroArch and Spectacol on AndroidUnix, Windows, MacOS X
ZX UlaxNice emulator with an interesting extra feature - specific games have been adapted by the author to run in full colourWindows, Linux with persuasion
RVMFairly new, easy to use, nice presentation and well documented (except the assembler) MacOS X, Linux, Windows
RetroArchIt runs software for many computers / consoles / arcade machines. Extensive debugger, more display and controller options than your brain can cope with, but many nice features like remembering keyboard / joystick configs per game. Also handy if you want to try games from many retro platforms with one tool. Can be a bit overwhelming to setup Windows, MacOS X, Linux, Raspberry, Android, IOS as well as on game consoles like PS2, PS3, PSP, PS Vita, Wii, Wii U, 2DS, 3DS, Switch, and more



Android

Run Spectrum games on your phone ! Or you can buy Android boxes / sticks around £30 and emulate Spectrum software on your TV. Note I'm not aware of any Spectrum development software on Android (but see ASM80 above)

Top options would be Retroarch if you want to run many platforms, or Spectacol or Xpectroid if you are mainly interested in the Spectrum, available in the Play Store

IOS / IPhone

More limited options here but there is Retroarch (with somewhat fiddly installation) or Spectaculator - with specific games available for free and some others for purchase - games that would be free on other platforms.

Or use the I Phone specific browser emulator https://axmasoft.com/ispeccy/
Other browser emulators may work fine

Raspberry PI

As with an Android box, you can buy a Raspberry PI very cheap with plenty of power to run Spectrum games. 

An additional option with Raspberry is to run "Bare Emulator"  - the emulator runs as a dedicated Spectrum without an operating system overhead. Or you can use Emulation Station / RetroArch to emulate many systems including a Spectrum

FGPA

Field Gate Programmable Arrays - the connoisseur's retro kit ; emulation usually involves some compromises, FGPA is hardware designed to run other hardware platforms very precisely - especially where it comes to timing - to minimise the lag between controller sound and display that can occur with normal software emulation. Technically the Spectrum Next is an FGPA machine but it's listed under original hardware. Expect to pay £200 - £400 for an FGPA machine.