Time to review the software
So, the code should be stuck in a loop doing this:
DBUG: LD HL,SIGNON1 ; Sign-on message
CALL PRINT ; Output string
JP DBUG
PRINT is coded thus:
PRINT: LD A,(HL) ; Get character
OR A ; Is it $00 ?
RET Z ; Then RETurn on terminator
RST 08H ; Print it
INC HL ; Next Character
JR PRINT ; Continue until $00
The software interrupt 08 jumps to the transmit routine which looks like this:
TXA: PUSH AF ; Store character
conout1: IN A,($80) ; Status byte
BIT 1,A ; Set Zero flag if still transmitting character
JR Z,conout1 ; Loop until flag signals ready
POP AF ; Retrieve character
OUT ($81),A ; Output the character
RET
As you are not seeing any writes I would guess that it is reading a 0 for the status bit and sitting in the tight loop jumping back to conout1 repeatedly reading the port, rather than executing the longer loop around outputting a whole string - you should be able to follow the loop on your 'scope.
The first thing I would check is that the data bus is connected correctly to the 68B50 - D0 from the Z80 going to D0 on the UART etc.
EDIT:
The 2nd thing I'd check is that the address lines are correctly connected to the chip select pins and the register select line RS
2nd EDIT:
It's also worth considering whether the CPU is executing the code that we think it is. The best tool would be a logic analyser but as we don't have one the 'scope can be used - trigger on MREQ and get a steady trace on CH1, then go over the other signals with CH2 to see what is happening.
In this case it does look as though we are running the read port/test bit/jump loop.
IN A,($80) consists of an M1 cycle (4 clocks), a memory read (for the port number) which is 3 clocks and then the I/O operation (4 clocks), 11 clocks in all.
BIT 1, A consists of 2 M1 cycles (8 clocks).
JR consists of 1 M1 cycle (4 clocks), a memory read (for the offset) which is 3 clocks, then the jump execution, 5 clocks, if the jump is taken.
So, you should get the following over 31 clock cycles, forgive the bad ASCII art
| IN A,($80) | BIT 1,A | JR conout1
| M1 cycle | Mem RD | IO cycle | M1 cycle | M1 cycle | M1 cycle | Mem RD | Jump
CLK |`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|_|`|
M1 |______|````````````````````````````````|_______|```````````|_______|```````````|_______|``````````````````````````````````````
So, basically M1 goes low, then there is a bit of a pause, then three M1 low pulses in a row, then a bit of a pause again and then the cycle repeats.
Happily the trace that you showed of M1 seems to show the above pattern so hopefully it is running the conout1 loop and you've just got a bit of a miswire somewhere.
Otherwise the 68B50 has stuck not transmitting for some reason.