Arduino UNO SPI Serial Clock (Pin 13 SCK) gives no output, but Blink Example (Pin 13 LED) works
I can't reproduce that. Using your first test code:

#include &lt;SPI.h&gt;

void setup() {
  //pinMode(13, OUTPUT);
  SPI.begin();

}

void loop() {
  SPI.transfer(0x00);  
}


I get this on the logic analyzer:



Also on my scope on pin 13:



As expected, a 4 MHz signal with gaps between the bytes.




  I expected the clock to be high frequency and continuous for SPI.


No, you get the clock when data is being transmitted and not at other times.


  Although likely irrelevant to the question, I noticed some periodic 6us gaps every ~1ms on this UNO Pin 10 (MOSI) signal. 


That would be the Timer 0 interrupt kicking in (roughly every 1 ms) and thus interrupting your looping code for a few microseconds.



Possible solution

Try enabling slave select (write LOW to pin 10). I think some versions of the logic analyzer ignore incoming data if the slave isn't selected.

In fact, pin 10 should be an output, otherwise the SPI hardware goes into slave mode. Some versions of the library might not set it to output.