The following is a brief record of things that go wrong with ZX81/TS1000 computers and how to recognize and fix them.
Your first clue as to what’s wrong is what does or doesn’t come up on the screen. If nothing comes up on the screen, check to see that the TV and computer have the same channel selected. Suspect a bad power adapter if channel selection is OK. Another possibility is the RF modulator (the shiny box inside the computer), but I have yet to see one of these fail.
To take the computer apart, remove the 3 to 5 screws from the bottom and separate the top and bottom halves. Two of the screws are under rubber feet. You have to do this if you want to fix it. The top half will have the computer printed circuit board attached. The first thing to check is the shiny grounding strip on the bottom of the circuit board. Sometimes one end (the end near the edge connection) breaks away from the circuit board. If so, reposition it and resolder it to the board. In nearly all repairs you will have access the top of the circuit board. Do this by taking the screws (usually 3) from the bottom of the circuit board. After removing the, slowly separate the case top from the circuit board being careful not to damage the keyboard ribbon connections. At this point you have two choices on how to proceed. If you pull the two keyboard ribbon cables from their connectors, you risk damage. You may (1) tear them or (2) not be able to reinsert them properly with out damaging the ends. The risk is either damaging the connections or damaging the connections. One alternative is to set the board and case top on their edges. This leaves a small space between the halves to get at the components on the board. However, you also risk damaging the ribbon connections this way too. This is one reason Budget Robotics sells replacement membrane keyboards.
Now, what will need fixing? If you have damaged keyboard ribbon connections (they get brittle from the heat sometimes), replace the keyboard. However, a common mistake in diagnosing a computer problem is to assume that if the keyboard doesn’t respond, the keyboard is bad. If a group of keys in one area of the keyboard are the only ones that don’t work, it’s a bad keyboard. Replace IC1 if none of the keys work (and the connection is OK) or if keys only fail in a shifted mode. Integrated circuit chip number 1 is one of the two largest chips and always socketed. In fact, IC1 is the most common component [illegible] fail.
The CPU (the other large one, marked Z80 on top) is the second most susceptible to failure.
The other IC chips on the board rarely fail. These chips a[illegible] the ROM (read only memory chip with the operating system inside) and the RAM (random access memory chip). Some machines have one, and some two RAM chips.
That is a corporate dump of the repair experience we have, good luck in you efforts.