The issue is complicated. In general it depends on the specific spinal reflex and on the specific anaesthetic. But it seems that the modern general anaesthetics usually do NOT block monosynaptic spinal reflexes (such as knee-jerk).
Here are two figures from Baars et al. 2009 that show that propofol and sevoflurane strongly inhibit withdrawal reflex (spinal two-synaptic reflex) but have almost no effect on the H-reflex (spinal monosynaptic reflex closely connected to the stretch reflex):
See below for many more details. Disclaimer: I know very little about anaesthesia and am simply writing down what I found after a couple of days searching and reading.
As @AliceD wrote, withdrawal reflex (a two-synaptic spinal cord reflex) usually disappears under any anaesthesia and this disappearance is used as a clinical sign of anaesthesia being deep enough for the surgery. As an example, this handout on monitoring anaesthetic depth in rodents has the following entry on the list of criteria for surgical anesthesia:
Withdrawal reflexes are absent! Try at least 2 toes and the ears so that you are sure that this reflex is absent.
Knee-jerk reflex: early history
On the other hand, it is known from the XIX century that stretch reflexes such as knee-jerk (note that they are monosynaptic) are very persistent. They can be abolished under some drugs (such as e.g. chloroform and ether) that act directly on the spine, but keep working under others. I am quoting from three old papers on the topic (here and below emphasis is mine).
Horsley, 1883, Note on the patellar knee-jerk:
In 1881, while experimenting (on myself) with this gas [nitrous oxide] for
a different purpose, it occurred to me to contrast the conditions of the superficial and deep "reflexes," taking the
plantar reflex as an example of the former, and the patellar
phenomenon to illustrate the latter. Being aware of the fact
that deep chloroform-narcosis abolishes all the "reflexes,"
superficial and deep, and yet that with ether-narcosis, however deep, ankle-clonus frequently, appears, I was scarcely
surprised to find that the knee-jerk persisted in the deepest
anaesthesia from nitrous oxide, while the superficial reflexes
Sherrington, 1892, Notes on the arrangement of some motor fibres in the lumbo-sacral plexus (p 671):
With regard to the effect of anaesthetics on the "jerk," a better proof of
the action of chloroform being rapid and direct on the nervous mechanisms of
the cord itself could hardly be found than in the speedy abolition of the
"jerk" by chloroform inhalation when the cord has been previously divided
in the thoracic region.
Simpson & Herring, 1905, The effect of cold narcosis on reflex action in warm-blooded animals:
The knee-jerk is a very persistent phenomenon. It is known to disappear in deep anesthesia produced by ether or chloroform [Sherrington 1892]. [...] The knee-jerk is as a rule easily elicited, and we find it present when
all the reflexes have disappeared. [...] The lower the
temperature the smaller the jerk. [...] In several animals that died at a low temperature the disappearance of the knee-jerk was simultaneous with death [...] The lowest temperature at which we found the knee-jerk present was 15° C., and this
was also the lowest temperature at which any of our animals were
still alive. [...] Pinching the skin does not provoke any response till the temperature is 25° C. [...] We have compared these results with similar ones obtained from
cats in recovery from deep ether anesthesia and find that the order of
reappearance agrees very closely.
Modern studies in animals
Hara & Harris, 2002, The Anesthetic Mechanism of Urethane: The Effects on
Neurotransmitter-Gated Ion Channels mention in the abstract that
Urethane is widely used as an anesthetic for animal studies because of its minimal effects on cardiovascular and respiratory systems and maintenance of spinal reflexes.
(they don't specify which reflexes specifically).
Ho & Waite, 2002, Effects of Different Anesthetics on the Paired-Pulse Depression of the H Reflex in Adult Rat study the effect of five different anaesthetics (ketamine, halothane, etomidate, saffan, and nembutal) on the Hoffman reflex (H-reflex), a monosynaptic reflex similar to the stretch reflex. They report that it keeps working under all five drugs, but the drugs modulate the decrease of the reflex under repeated stimulation (this decrease depends on the descending controls and is affected differently by different drugs):
The results suggest a preferential
action of some anesthetics on descending pathways
involved in reflex modulation [...]
Strangely, Ho & Waite do not describe how anaesthesia affects the reflex strength following an unrepeated stimulation, but in any case it is clear the the reflex keeps functioning.
What about the anaesthetics widely used in humans, such as e.g. propofol? Matute et al., 2004, Effects of propofol and sevoflurane on the excitability of rat spinal motoneurones and nociceptive reflexes in vitro:
We used an isolated spinal cord in vitro preparation from rat pups and superfused the anaesthetics at known concentrations. [...] Applied at anaesthetic concentrations, [...] sevoflurane produced a large depressant effect on the monosynaptic reflex whereas propofol was ineffective. [...] Sevoflurane produces large inhibitory effects on nociceptive and non-nociceptive reflexes which are likely to contribute to immobility during surgery. Compared with sevoflurane, propofol appears to have much weaker effects on spinal reflexes such as those recorded in an isolated preparation.
Kerz et al., 2001, Effects of Propofol on H-reflex in Humans:
Previous studies of motoneuron excitability using H-reflexes or F-waves during general anesthesia found significant depression after administration of halothane, enflurane, isoflurane, desflurane, or nitrous oxide. [...]
Recommended propofol doses for induction and maintenance only had a transient effect on the H-reflex and were no longer demonstrable after 10 min of propofol anesthesia. [...] Immobility during propofol anesthesia [...] does not seem to be caused by a depression of spinal motoneuron circuit excitability.
So why is then the withdrawal reflex so easily abolished and the stretch reflex is so persistent (e.g. under propofol)? This does not seem to be well understood. Baars et al., 2009, Effects of Sevoflurane and Propofol on the Nociceptive Withdrawal Reflex and on the H Reflex write
Both H reflex and nociceptive withdrawal reflexes are reduced dose-dependently by propofol and sevoflurane in humans. [...] We have shown a striking difference of the degree to which both reflexes are reduced by the two anesthetics. The relative reduction of the polysynaptic RIII reflex amplitude was more than four times higher than that of the H reflex. This difference can probably be attributed to the higher number of neurons interposed in the RIII reflex pathway in comparison to the H reflex.
Note that the last sentence is very vague; it says that the difference is "probably" due to monosynaptic vs. polysynaptic mechanisms of these two reflexes but does not offer any explanation. Withdrawal reflex is mediated by an excitatory interneuron and propofol acts on GABA (which is inhibitory transmitter), so it remains unclear. This study is from 2009 and I did not find anything relevant in the papers that cite it.