Robert:

I don't see any reason why you can't check the generator output by
connecting it to a scope (to monitor the waveform and/or voltage),
subject to the following (in the US, at least).  I assume that your
generator is producing single phase power, either 120 VAC single phase
or 120/240 VAC single phase:

1.  To begin with, there's no reason why you can't connect the
generator's ground to your house ground.  But do not make any other
connections from generator wiring to house wiring.  The ground (the
National Electrical Code "grounding conductor") will generally be
identified as a bare, or green insulated, or green with yellow stripe
insulated, conductor.  Generator (and house) current carrying conductors
will be either neutral conductors (National Electrical Code "grounded
conductor"), which are generally colored white or grey, or phase
conductors (colored differently from the assigned colors for grounded
and grounding conductors.)

2.  Power up the scope from "house" power, using a probe with a
sufficient voltage rating (many 10x probes are rated to 600 V).  You
will need to make a grounding connection from the scope ground (scope
chassis) to the generator ground; this can be either an individual
"stand alone" conductor, or the shield/braid of the scope probe.  I
recommend a separate conductor rather than the scope probe shield; an
alligator clip jumper cable should be sufficient.  (Note: beware of
cheap alligator clip jumper cables with crimped connections at the
alligator clips; the connections should be soldered.)

The US standard NEMA 5-15R (15 amp) and 5-20R (20 amp) 3-terminal
outlets have the grounding terminal sized to accept a 3/16" diameter
metal rod; I've successfully used pieces of round copper or brass rod to
provide a good grounding stud, and then used alligator clip leads to
extend the ground connection.

3. Provided that you've got the ground jumper in place, you should be
able to probe the generator's output connector individual pins with the
scope probe's tip.  And you won't need to connect the scope probe's
grounding clip to the generator's frame (as this connection will be
accomplished by the alligator clip jumper cable described above.  If
probing a 120 VAC outlet (3 terminal), you should wind up reading 0.0
volts when probing the ground pin, and 0.0 volts when probing the
neutral pin, and ~170 VAC peak (340 volts peak-to-peak) when probing the
"hot" terminal of the 120 VAC outlet.  Of course, make sure that you've
got the scope set up with a "less sensitive" vertical deflection; 5
V/div should be adequate if you are using a 10X probe.

I know that there have been quite a few threads on TekScopes dealing
with possible safety issues when using isolation transformers, or
autotransformers (e.g. Variacs) that "carry thru" a ground connection
when testing switch mode power supplies, but I don't see any of these
issues applying for what you are describing.  But I've been wrong
before, so I hope others will correct me if I'm wrong on this issue.

Mike Dinolfo N4MWP